3.6.43 \(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^6} \, dx\) [543]

Optimal. Leaf size=195 \[ -\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {3 a^2 c^3 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}} \]

[Out]

-1/4*c*d*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^2/(e*x+d)^4-1/5*e*(c*x^2+a)^(5/2)/(a*e^2+c*d^2)/(e*x+d)^5-
3/8*a^2*c^3*d*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(7/2)-3/8*a*c^2*d*(-c*d*
x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)^2

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Rubi [A]
time = 0.18, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {745, 735, 739, 212} \begin {gather*} -\frac {3 a^2 c^3 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac {3 a c^2 d \sqrt {a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {c d \left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(-3*a*c^2*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (c*d*(a*e - c*d*x)*(a + c*x^2)^
(3/2))/(4*(c*d^2 + a*e^2)^2*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(5*(c*d^2 + a*e^2)*(d + e*x)^5) - (3*a^2*c^3*
d*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(-2*a*e + (2*c
*d)*x)*((a + c*x^2)^p/(2*(m + 1)*(c*d^2 + a*e^2))), x] - Dist[4*a*c*(p/(2*(m + 1)*(c*d^2 + a*e^2))), Int[(d +
e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2
, 0] && GtQ[p, 0]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx &=-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac {(c d) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{c d^2+a e^2}\\ &=-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac {\left (3 a c^2 d\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{4 \left (c d^2+a e^2\right )^2}\\ &=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac {\left (3 a^2 c^3 d\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {\left (3 a^2 c^3 d\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {3 a^2 c^3 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 2.70, size = 253, normalized size = 1.30 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-8 a^4 e^5+2 c^4 d^4 x^3 (5 d+e x)-2 a^3 c e^3 \left (13 d^2+5 d e x+8 e^2 x^2\right )+a c^3 d^2 x \left (25 d^3+29 d^2 e x+45 d e^2 x^2+9 e^3 x^3\right )-a^2 c^2 e \left (33 d^4+45 d^3 e x+77 d^2 e^2 x^2+25 d e^3 x^3+8 e^4 x^4\right )\right )}{40 \left (c d^2+a e^2\right )^3 (d+e x)^5}+\frac {3 a^2 c^3 d \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{4 \left (-c d^2-a e^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(Sqrt[a + c*x^2]*(-8*a^4*e^5 + 2*c^4*d^4*x^3*(5*d + e*x) - 2*a^3*c*e^3*(13*d^2 + 5*d*e*x + 8*e^2*x^2) + a*c^3*
d^2*x*(25*d^3 + 29*d^2*e*x + 45*d*e^2*x^2 + 9*e^3*x^3) - a^2*c^2*e*(33*d^4 + 45*d^3*e*x + 77*d^2*e^2*x^2 + 25*
d*e^3*x^3 + 8*e^4*x^4)))/(40*(c*d^2 + a*e^2)^3*(d + e*x)^5) + (3*a^2*c^3*d*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[
a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(4*(-(c*d^2) - a*e^2)^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4107\) vs. \(2(175)=350\).
time = 0.45, size = 4108, normalized size = 21.07

method	result	size
default	\(\text {Expression too large to display}\)	\(4108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

1/e^6*(-1/5/(a*e^2+c*d^2)*e^2/(x+d/e)^5*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)+c*d*e/(a*e^2+c*d
^2)*(-1/4/(a*e^2+c*d^2)*e^2/(x+d/e)^4*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)+3/4*c*d*e/(a*e^2+c
*d^2)*(-1/3/(a*e^2+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)+1/3*c*d*e/(a*e^2
+c*d^2)*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)-1/2*c*d*e/(a*e
^2+c*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)-3*c*d*e/(a*e^2+c
*d^2)*(1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d/e*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)
^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+
d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))+(a*e^2+c*d^2)/e^2*((c*(x+d/e)^2-2*c*d/e*
(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2
+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*
e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))))+4*c/(a*e^2+c*d^2)*e^2*
(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2+c*d^2)/e^2
-4*c^2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*
(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d
^2)/e^2)^(1/2)))))+3/2*c/(a*e^2+c*d^2)*e^2*(1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d/e*(1
/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*
c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))+(a*
e^2+c*d^2)/e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2
)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*
e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/
2))/(x+d/e)))))+2/3*c/(a*e^2+c*d^2)*e^2*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^
2)/e^2)^(5/2)-3*c*d*e/(a*e^2+c*d^2)*(1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d/e*(1/4*(2*c
*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2
/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))+(a*e^2+c*d
^2)/e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x
+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d
^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+
d/e))))+4*c/(a*e^2+c*d^2)*e^2*(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/
2)+3/16*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e
^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e
)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))))))+1/4*c/(a*e^2+c*d^2)*e^2*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^2*(c
*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)-1/2*c*d*e/(a*e^2+c*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(
x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)-3*c*d*e/(a*e^2+c*d^2)*(1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e
^2+c*d^2)/e^2)^(3/2)-c*d/e*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+
1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(
a*e^2+c*d^2)/e^2)^(1/2)))+(a*e^2+c*d^2)/e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e
*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^
2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(
x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))))+4*c/(a*e^2+c*d^2)*e^2*(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*
c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)
/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln(
(-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))))+3/2*c/(a*e^2+c*d^2)*e^2*(
1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d/e*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*
d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c
^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))+(a*e^2+c*d^2)/e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)
+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/...

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1680 vs. \(2 (178) = 356\).
time = 0.39, size = 1680, normalized size = 8.62 \begin {gather*} \frac {3 \, \sqrt {c x^{2} + a} c^{5} d^{5}}{8 \, {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )}} - \frac {3 \, \sqrt {c x^{2} + a} c^{5} d^{4} x}{8 \, {\left (c^{4} d^{8} e^{2} + 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} + 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{4} d^{4}}{8 \, {\left (c^{4} d^{8} x e^{2} + c^{4} d^{9} e + 4 \, a c^{3} d^{6} x e^{4} + 4 \, a c^{3} d^{7} e^{3} + 6 \, a^{2} c^{2} d^{4} x e^{6} + 6 \, a^{2} c^{2} d^{5} e^{5} + 4 \, a^{3} c d^{2} x e^{8} + 4 \, a^{3} c d^{3} e^{7} + a^{4} x e^{10} + a^{4} d e^{9}\right )}} + \frac {3 \, c^{5} d^{5} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-11\right )}}{8 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {7}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{3} d^{3}}{8 \, {\left (c^{4} d^{8} x^{2} e + c^{4} d^{10} e^{\left (-1\right )} + 2 \, c^{4} d^{9} x + 4 \, a c^{3} d^{6} x^{2} e^{3} + 8 \, a c^{3} d^{7} x e^{2} + 4 \, a c^{3} d^{8} e + 6 \, a^{2} c^{2} d^{4} x^{2} e^{5} + 12 \, a^{2} c^{2} d^{5} x e^{4} + 6 \, a^{2} c^{2} d^{6} e^{3} + 4 \, a^{3} c d^{2} x^{2} e^{7} + 8 \, a^{3} c d^{3} x e^{6} + 4 \, a^{3} c d^{4} e^{5} + a^{4} x^{2} e^{9} + 2 \, a^{4} d x e^{8} + a^{4} d^{2} e^{7}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{4} d^{3}}{8 \, {\left (c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )}} - \frac {3 \, \sqrt {c x^{2} + a} c^{4} d^{3}}{4 \, {\left (c^{3} d^{6} e^{3} + 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )}} + \frac {3 \, \sqrt {c x^{2} + a} c^{4} d^{2} x}{8 \, {\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )}} - \frac {3 \, c^{4} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-9\right )}}{4 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {5}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{2} d^{2}}{4 \, {\left (c^{3} d^{6} x^{3} e^{2} + 3 \, c^{3} d^{7} x^{2} e + c^{3} d^{9} e^{\left (-1\right )} + 3 \, c^{3} d^{8} x + 3 \, a c^{2} d^{4} x^{3} e^{4} + 9 \, a c^{2} d^{5} x^{2} e^{3} + 9 \, a c^{2} d^{6} x e^{2} + 3 \, a c^{2} d^{7} e + 3 \, a^{2} c d^{2} x^{3} e^{6} + 9 \, a^{2} c d^{3} x^{2} e^{5} + 9 \, a^{2} c d^{4} x e^{4} + 3 \, a^{2} c d^{5} e^{3} + a^{3} x^{3} e^{8} + 3 \, a^{3} d x^{2} e^{7} + 3 \, a^{3} d^{2} x e^{6} + a^{3} d^{3} e^{5}\right )}} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3} d^{2}}{8 \, {\left (c^{3} d^{6} x e^{2} + c^{3} d^{7} e + 3 \, a c^{2} d^{4} x e^{4} + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{2} x e^{6} + 3 \, a^{2} c d^{3} e^{5} + a^{3} x e^{8} + a^{3} d e^{7}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{2} d}{8 \, {\left (c^{3} d^{6} x^{2} e + c^{3} d^{8} e^{\left (-1\right )} + 2 \, c^{3} d^{7} x + 3 \, a c^{2} d^{4} x^{2} e^{3} + 6 \, a c^{2} d^{5} x e^{2} + 3 \, a c^{2} d^{6} e + 3 \, a^{2} c d^{2} x^{2} e^{5} + 6 \, a^{2} c d^{3} x e^{4} + 3 \, a^{2} c d^{4} e^{3} + a^{3} x^{2} e^{7} + 2 \, a^{3} d x e^{6} + a^{3} d^{2} e^{5}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3} d}{8 \, {\left (c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )}} + \frac {3 \, c^{3} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-7\right )}}{8 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c d}{4 \, {\left (c^{2} d^{4} x^{4} e^{3} + 4 \, c^{2} d^{5} x^{3} e^{2} + 6 \, c^{2} d^{6} x^{2} e + c^{2} d^{8} e^{\left (-1\right )} + 4 \, c^{2} d^{7} x + 2 \, a c d^{2} x^{4} e^{5} + 8 \, a c d^{3} x^{3} e^{4} + 12 \, a c d^{4} x^{2} e^{3} + 8 \, a c d^{5} x e^{2} + 2 \, a c d^{6} e + a^{2} x^{4} e^{7} + 4 \, a^{2} d x^{3} e^{6} + 6 \, a^{2} d^{2} x^{2} e^{5} + 4 \, a^{2} d^{3} x e^{4} + a^{2} d^{4} e^{3}\right )}} + \frac {3 \, \sqrt {c x^{2} + a} c^{3} d}{8 \, {\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{5 \, {\left (c d^{2} x^{5} e^{4} + 5 \, c d^{3} x^{4} e^{3} + 10 \, c d^{4} x^{3} e^{2} + 10 \, c d^{5} x^{2} e + c d^{7} e^{\left (-1\right )} + 5 \, c d^{6} x + a x^{5} e^{6} + 5 \, a d x^{4} e^{5} + 10 \, a d^{2} x^{3} e^{4} + 10 \, a d^{3} x^{2} e^{3} + 5 \, a d^{4} x e^{2} + a d^{5} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

3/8*sqrt(c*x^2 + a)*c^5*d^5/(c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^3*c*d^2*e^9 + a^4*e^11) -
3/8*sqrt(c*x^2 + a)*c^5*d^4*x/(c^4*d^8*e^2 + 4*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 + 4*a^3*c*d^2*e^8 + a^4*e^10
) + 1/8*(c*x^2 + a)^(3/2)*c^4*d^4/(c^4*d^8*x*e^2 + c^4*d^9*e + 4*a*c^3*d^6*x*e^4 + 4*a*c^3*d^7*e^3 + 6*a^2*c^2
*d^4*x*e^6 + 6*a^2*c^2*d^5*e^5 + 4*a^3*c*d^2*x*e^8 + 4*a^3*c*d^3*e^7 + a^4*x*e^10 + a^4*d*e^9) + 3/8*c^5*d^5*a
rcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-11)/(c*d^2*e^(-2) + a)^(7/2) - 1/8*(
c*x^2 + a)^(5/2)*c^3*d^3/(c^4*d^8*x^2*e + c^4*d^10*e^(-1) + 2*c^4*d^9*x + 4*a*c^3*d^6*x^2*e^3 + 8*a*c^3*d^7*x*
e^2 + 4*a*c^3*d^8*e + 6*a^2*c^2*d^4*x^2*e^5 + 12*a^2*c^2*d^5*x*e^4 + 6*a^2*c^2*d^6*e^3 + 4*a^3*c*d^2*x^2*e^7 +
8*a^3*c*d^3*x*e^6 + 4*a^3*c*d^4*e^5 + a^4*x^2*e^9 + 2*a^4*d*x*e^8 + a^4*d^2*e^7) + 1/8*(c*x^2 + a)^(3/2)*c^4*
d^3/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) - 3/4*sqrt(c*x^2 + a)*c^4*d^
3/(c^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3*e^9) + 3/8*sqrt(c*x^2 + a)*c^4*d^2*x/(c^3*d^6*e^2 + 3
*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 + a^3*e^8) - 3/4*c^4*d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a
*c)*abs(x*e + d)))*e^(-9)/(c*d^2*e^(-2) + a)^(5/2) - 1/4*(c*x^2 + a)^(5/2)*c^2*d^2/(c^3*d^6*x^3*e^2 + 3*c^3*d^
7*x^2*e + c^3*d^9*e^(-1) + 3*c^3*d^8*x + 3*a*c^2*d^4*x^3*e^4 + 9*a*c^2*d^5*x^2*e^3 + 9*a*c^2*d^6*x*e^2 + 3*a*c
^2*d^7*e + 3*a^2*c*d^2*x^3*e^6 + 9*a^2*c*d^3*x^2*e^5 + 9*a^2*c*d^4*x*e^4 + 3*a^2*c*d^5*e^3 + a^3*x^3*e^8 + 3*a
^3*d*x^2*e^7 + 3*a^3*d^2*x*e^6 + a^3*d^3*e^5) - 3/8*(c*x^2 + a)^(3/2)*c^3*d^2/(c^3*d^6*x*e^2 + c^3*d^7*e + 3*a
*c^2*d^4*x*e^4 + 3*a*c^2*d^5*e^3 + 3*a^2*c*d^2*x*e^6 + 3*a^2*c*d^3*e^5 + a^3*x*e^8 + a^3*d*e^7) - 1/8*(c*x^2 +
a)^(5/2)*c^2*d/(c^3*d^6*x^2*e + c^3*d^8*e^(-1) + 2*c^3*d^7*x + 3*a*c^2*d^4*x^2*e^3 + 6*a*c^2*d^5*x*e^2 + 3*a*
c^2*d^6*e + 3*a^2*c*d^2*x^2*e^5 + 6*a^2*c*d^3*x*e^4 + 3*a^2*c*d^4*e^3 + a^3*x^2*e^7 + 2*a^3*d*x*e^6 + a^3*d^2*
e^5) + 1/8*(c*x^2 + a)^(3/2)*c^3*d/(c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7) + 3/8*c^3*d*arcsi
nh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-7)/(c*d^2*e^(-2) + a)^(3/2) - 1/4*(c*x^2
+ a)^(5/2)*c*d/(c^2*d^4*x^4*e^3 + 4*c^2*d^5*x^3*e^2 + 6*c^2*d^6*x^2*e + c^2*d^8*e^(-1) + 4*c^2*d^7*x + 2*a*c*
d^2*x^4*e^5 + 8*a*c*d^3*x^3*e^4 + 12*a*c*d^4*x^2*e^3 + 8*a*c*d^5*x*e^2 + 2*a*c*d^6*e + a^2*x^4*e^7 + 4*a^2*d*x
^3*e^6 + 6*a^2*d^2*x^2*e^5 + 4*a^2*d^3*x*e^4 + a^2*d^4*e^3) + 3/8*sqrt(c*x^2 + a)*c^3*d/(c^2*d^4*e^3 + 2*a*c*d
^2*e^5 + a^2*e^7) - 1/5*(c*x^2 + a)^(5/2)/(c*d^2*x^5*e^4 + 5*c*d^3*x^4*e^3 + 10*c*d^4*x^3*e^2 + 10*c*d^5*x^2*e
+ c*d^7*e^(-1) + 5*c*d^6*x + a*x^5*e^6 + 5*a*d*x^4*e^5 + 10*a*d^2*x^3*e^4 + 10*a*d^3*x^2*e^3 + 5*a*d^4*x*e^2
+ a*d^5*e)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (178) = 356\).
time = 7.93, size = 1653, normalized size = 8.48 \begin {gather*} \left [\frac {15 \, {\left (a^{2} c^{3} d x^{5} e^{5} + 5 \, a^{2} c^{3} d^{2} x^{4} e^{4} + 10 \, a^{2} c^{3} d^{3} x^{3} e^{3} + 10 \, a^{2} c^{3} d^{4} x^{2} e^{2} + 5 \, a^{2} c^{3} d^{5} x e + a^{2} c^{3} d^{6}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (10 \, c^{5} d^{7} x^{3} + 25 \, a c^{4} d^{7} x - 8 \, {\left (a^{3} c^{2} x^{4} + 2 \, a^{4} c x^{2} + a^{5}\right )} e^{7} - 5 \, {\left (5 \, a^{3} c^{2} d x^{3} + 2 \, a^{4} c d x\right )} e^{6} + {\left (a^{2} c^{3} d^{2} x^{4} - 93 \, a^{3} c^{2} d^{2} x^{2} - 34 \, a^{4} c d^{2}\right )} e^{5} + 5 \, {\left (4 \, a^{2} c^{3} d^{3} x^{3} - 11 \, a^{3} c^{2} d^{3} x\right )} e^{4} + {\left (11 \, a c^{4} d^{4} x^{4} - 48 \, a^{2} c^{3} d^{4} x^{2} - 59 \, a^{3} c^{2} d^{4}\right )} e^{3} + 5 \, {\left (11 \, a c^{4} d^{5} x^{3} - 4 \, a^{2} c^{3} d^{5} x\right )} e^{2} + {\left (2 \, c^{5} d^{6} x^{4} + 29 \, a c^{4} d^{6} x^{2} - 33 \, a^{2} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{80 \, {\left (5 \, c^{4} d^{12} x e + c^{4} d^{13} + a^{4} x^{5} e^{13} + 5 \, a^{4} d x^{4} e^{12} + 2 \, {\left (2 \, a^{3} c d^{2} x^{5} + 5 \, a^{4} d^{2} x^{3}\right )} e^{11} + 10 \, {\left (2 \, a^{3} c d^{3} x^{4} + a^{4} d^{3} x^{2}\right )} e^{10} + {\left (6 \, a^{2} c^{2} d^{4} x^{5} + 40 \, a^{3} c d^{4} x^{3} + 5 \, a^{4} d^{4} x\right )} e^{9} + {\left (30 \, a^{2} c^{2} d^{5} x^{4} + 40 \, a^{3} c d^{5} x^{2} + a^{4} d^{5}\right )} e^{8} + 4 \, {\left (a c^{3} d^{6} x^{5} + 15 \, a^{2} c^{2} d^{6} x^{3} + 5 \, a^{3} c d^{6} x\right )} e^{7} + 4 \, {\left (5 \, a c^{3} d^{7} x^{4} + 15 \, a^{2} c^{2} d^{7} x^{2} + a^{3} c d^{7}\right )} e^{6} + {\left (c^{4} d^{8} x^{5} + 40 \, a c^{3} d^{8} x^{3} + 30 \, a^{2} c^{2} d^{8} x\right )} e^{5} + {\left (5 \, c^{4} d^{9} x^{4} + 40 \, a c^{3} d^{9} x^{2} + 6 \, a^{2} c^{2} d^{9}\right )} e^{4} + 10 \, {\left (c^{4} d^{10} x^{3} + 2 \, a c^{3} d^{10} x\right )} e^{3} + 2 \, {\left (5 \, c^{4} d^{11} x^{2} + 2 \, a c^{3} d^{11}\right )} e^{2}\right )}}, \frac {15 \, {\left (a^{2} c^{3} d x^{5} e^{5} + 5 \, a^{2} c^{3} d^{2} x^{4} e^{4} + 10 \, a^{2} c^{3} d^{3} x^{3} e^{3} + 10 \, a^{2} c^{3} d^{4} x^{2} e^{2} + 5 \, a^{2} c^{3} d^{5} x e + a^{2} c^{3} d^{6}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + {\left (10 \, c^{5} d^{7} x^{3} + 25 \, a c^{4} d^{7} x - 8 \, {\left (a^{3} c^{2} x^{4} + 2 \, a^{4} c x^{2} + a^{5}\right )} e^{7} - 5 \, {\left (5 \, a^{3} c^{2} d x^{3} + 2 \, a^{4} c d x\right )} e^{6} + {\left (a^{2} c^{3} d^{2} x^{4} - 93 \, a^{3} c^{2} d^{2} x^{2} - 34 \, a^{4} c d^{2}\right )} e^{5} + 5 \, {\left (4 \, a^{2} c^{3} d^{3} x^{3} - 11 \, a^{3} c^{2} d^{3} x\right )} e^{4} + {\left (11 \, a c^{4} d^{4} x^{4} - 48 \, a^{2} c^{3} d^{4} x^{2} - 59 \, a^{3} c^{2} d^{4}\right )} e^{3} + 5 \, {\left (11 \, a c^{4} d^{5} x^{3} - 4 \, a^{2} c^{3} d^{5} x\right )} e^{2} + {\left (2 \, c^{5} d^{6} x^{4} + 29 \, a c^{4} d^{6} x^{2} - 33 \, a^{2} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{40 \, {\left (5 \, c^{4} d^{12} x e + c^{4} d^{13} + a^{4} x^{5} e^{13} + 5 \, a^{4} d x^{4} e^{12} + 2 \, {\left (2 \, a^{3} c d^{2} x^{5} + 5 \, a^{4} d^{2} x^{3}\right )} e^{11} + 10 \, {\left (2 \, a^{3} c d^{3} x^{4} + a^{4} d^{3} x^{2}\right )} e^{10} + {\left (6 \, a^{2} c^{2} d^{4} x^{5} + 40 \, a^{3} c d^{4} x^{3} + 5 \, a^{4} d^{4} x\right )} e^{9} + {\left (30 \, a^{2} c^{2} d^{5} x^{4} + 40 \, a^{3} c d^{5} x^{2} + a^{4} d^{5}\right )} e^{8} + 4 \, {\left (a c^{3} d^{6} x^{5} + 15 \, a^{2} c^{2} d^{6} x^{3} + 5 \, a^{3} c d^{6} x\right )} e^{7} + 4 \, {\left (5 \, a c^{3} d^{7} x^{4} + 15 \, a^{2} c^{2} d^{7} x^{2} + a^{3} c d^{7}\right )} e^{6} + {\left (c^{4} d^{8} x^{5} + 40 \, a c^{3} d^{8} x^{3} + 30 \, a^{2} c^{2} d^{8} x\right )} e^{5} + {\left (5 \, c^{4} d^{9} x^{4} + 40 \, a c^{3} d^{9} x^{2} + 6 \, a^{2} c^{2} d^{9}\right )} e^{4} + 10 \, {\left (c^{4} d^{10} x^{3} + 2 \, a c^{3} d^{10} x\right )} e^{3} + 2 \, {\left (5 \, c^{4} d^{11} x^{2} + 2 \, a c^{3} d^{11}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

[1/80*(15*(a^2*c^3*d*x^5*e^5 + 5*a^2*c^3*d^2*x^4*e^4 + 10*a^2*c^3*d^3*x^3*e^3 + 10*a^2*c^3*d^4*x^2*e^2 + 5*a^2
*c^3*d^5*x*e + a^2*c^3*d^6)*sqrt(c*d^2 + a*e^2)*log(-(2*c^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 + 2*sqrt(c*d^2 + a
*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2*a^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(10*c^5*d^7*x^3 +
25*a*c^4*d^7*x - 8*(a^3*c^2*x^4 + 2*a^4*c*x^2 + a^5)*e^7 - 5*(5*a^3*c^2*d*x^3 + 2*a^4*c*d*x)*e^6 + (a^2*c^3*d^
2*x^4 - 93*a^3*c^2*d^2*x^2 - 34*a^4*c*d^2)*e^5 + 5*(4*a^2*c^3*d^3*x^3 - 11*a^3*c^2*d^3*x)*e^4 + (11*a*c^4*d^4*
x^4 - 48*a^2*c^3*d^4*x^2 - 59*a^3*c^2*d^4)*e^3 + 5*(11*a*c^4*d^5*x^3 - 4*a^2*c^3*d^5*x)*e^2 + (2*c^5*d^6*x^4 +
29*a*c^4*d^6*x^2 - 33*a^2*c^3*d^6)*e)*sqrt(c*x^2 + a))/(5*c^4*d^12*x*e + c^4*d^13 + a^4*x^5*e^13 + 5*a^4*d*x^
4*e^12 + 2*(2*a^3*c*d^2*x^5 + 5*a^4*d^2*x^3)*e^11 + 10*(2*a^3*c*d^3*x^4 + a^4*d^3*x^2)*e^10 + (6*a^2*c^2*d^4*x
^5 + 40*a^3*c*d^4*x^3 + 5*a^4*d^4*x)*e^9 + (30*a^2*c^2*d^5*x^4 + 40*a^3*c*d^5*x^2 + a^4*d^5)*e^8 + 4*(a*c^3*d^
6*x^5 + 15*a^2*c^2*d^6*x^3 + 5*a^3*c*d^6*x)*e^7 + 4*(5*a*c^3*d^7*x^4 + 15*a^2*c^2*d^7*x^2 + a^3*c*d^7)*e^6 + (
c^4*d^8*x^5 + 40*a*c^3*d^8*x^3 + 30*a^2*c^2*d^8*x)*e^5 + (5*c^4*d^9*x^4 + 40*a*c^3*d^9*x^2 + 6*a^2*c^2*d^9)*e^
4 + 10*(c^4*d^10*x^3 + 2*a*c^3*d^10*x)*e^3 + 2*(5*c^4*d^11*x^2 + 2*a*c^3*d^11)*e^2), 1/40*(15*(a^2*c^3*d*x^5*e
^5 + 5*a^2*c^3*d^2*x^4*e^4 + 10*a^2*c^3*d^3*x^3*e^3 + 10*a^2*c^3*d^4*x^2*e^2 + 5*a^2*c^3*d^5*x*e + a^2*c^3*d^6
)*sqrt(-c*d^2 - a*e^2)*arctan(-sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(c^2*d^2*x^2 + a*c*d^2 + (a*
c*x^2 + a^2)*e^2)) + (10*c^5*d^7*x^3 + 25*a*c^4*d^7*x - 8*(a^3*c^2*x^4 + 2*a^4*c*x^2 + a^5)*e^7 - 5*(5*a^3*c^2
*d*x^3 + 2*a^4*c*d*x)*e^6 + (a^2*c^3*d^2*x^4 - 93*a^3*c^2*d^2*x^2 - 34*a^4*c*d^2)*e^5 + 5*(4*a^2*c^3*d^3*x^3 -
11*a^3*c^2*d^3*x)*e^4 + (11*a*c^4*d^4*x^4 - 48*a^2*c^3*d^4*x^2 - 59*a^3*c^2*d^4)*e^3 + 5*(11*a*c^4*d^5*x^3 -
4*a^2*c^3*d^5*x)*e^2 + (2*c^5*d^6*x^4 + 29*a*c^4*d^6*x^2 - 33*a^2*c^3*d^6)*e)*sqrt(c*x^2 + a))/(5*c^4*d^12*x*e
+ c^4*d^13 + a^4*x^5*e^13 + 5*a^4*d*x^4*e^12 + 2*(2*a^3*c*d^2*x^5 + 5*a^4*d^2*x^3)*e^11 + 10*(2*a^3*c*d^3*x^4
+ a^4*d^3*x^2)*e^10 + (6*a^2*c^2*d^4*x^5 + 40*a^3*c*d^4*x^3 + 5*a^4*d^4*x)*e^9 + (30*a^2*c^2*d^5*x^4 + 40*a^3
*c*d^5*x^2 + a^4*d^5)*e^8 + 4*(a*c^3*d^6*x^5 + 15*a^2*c^2*d^6*x^3 + 5*a^3*c*d^6*x)*e^7 + 4*(5*a*c^3*d^7*x^4 +
15*a^2*c^2*d^7*x^2 + a^3*c*d^7)*e^6 + (c^4*d^8*x^5 + 40*a*c^3*d^8*x^3 + 30*a^2*c^2*d^8*x)*e^5 + (5*c^4*d^9*x^4
+ 40*a*c^3*d^9*x^2 + 6*a^2*c^2*d^9)*e^4 + 10*(c^4*d^10*x^3 + 2*a*c^3*d^10*x)*e^3 + 2*(5*c^4*d^11*x^2 + 2*a*c^
3*d^11)*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**6,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**6, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1246 vs. \(2 (178) = 356\).
time = 0.84, size = 1246, normalized size = 6.39 \begin {gather*} -\frac {3 \, a^{2} c^{3} d \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{4 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} c^{\frac {13}{2}} d^{8} e + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{7} d^{9} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} c^{6} d^{7} e^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} c^{\frac {11}{2}} d^{6} e^{3} - 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {13}{2}} d^{8} e - 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{6} d^{7} e^{2} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a c^{\frac {11}{2}} d^{6} e^{3} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a c^{5} d^{5} e^{4} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{6} d^{7} e^{2} + 120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} a c^{\frac {9}{2}} d^{4} e^{5} - 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {11}{2}} d^{6} e^{3} - 788 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{2} c^{5} d^{5} e^{4} - 530 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a^{2} c^{\frac {9}{2}} d^{4} e^{5} - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {11}{2}} d^{6} e^{3} - 230 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a^{2} c^{4} d^{3} e^{6} + 400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{3} c^{5} d^{5} e^{4} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} a^{2} c^{\frac {7}{2}} d^{2} e^{7} + 1170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{3} c^{\frac {9}{2}} d^{4} e^{5} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} a^{2} c^{3} d e^{8} + 910 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{3} c^{4} d^{3} e^{6} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{4} c^{5} d^{5} e^{4} + 570 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a^{3} c^{\frac {7}{2}} d^{2} e^{7} - 230 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{4} c^{\frac {9}{2}} d^{4} e^{5} + 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a^{3} c^{3} d e^{8} - 770 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{4} c^{4} d^{3} e^{6} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} a^{3} c^{\frac {5}{2}} e^{9} - 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{4} c^{\frac {7}{2}} d^{2} e^{7} - 2 \, a^{5} c^{\frac {9}{2}} d^{4} e^{5} - 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{4} c^{3} d e^{8} + 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{5} c^{4} d^{3} e^{6} + 350 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{5} c^{\frac {7}{2}} d^{2} e^{7} + 170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{5} c^{3} d e^{8} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{5} c^{\frac {5}{2}} e^{9} - 9 \, a^{6} c^{\frac {7}{2}} d^{2} e^{7} - 65 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{6} c^{3} d e^{8} + 8 \, a^{7} c^{\frac {5}{2}} e^{9}}{20 \, {\left (c^{3} d^{6} e^{4} + 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} + a^{3} e^{10}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

-3/4*a^2*c^3*d*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*
d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) + 1/20*(80*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)
*d^8*e + 32*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*d^9 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*d^7*e^2 + 40*(sqr
t(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*d^6*e^3 - 80*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*d^8*e - 16*(sqrt(
c)*x - sqrt(c*x^2 + a))^5*a*c^6*d^7*e^2 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*d^6*e^3 + 240*(sqrt(c
)*x - sqrt(c*x^2 + a))^7*a*c^5*d^5*e^4 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^6*d^7*e^2 + 120*(sqrt(c)*x -
sqrt(c*x^2 + a))^8*a*c^(9/2)*d^4*e^5 - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)*d^6*e^3 - 788*(sqrt(c
)*x - sqrt(c*x^2 + a))^5*a^2*c^5*d^5*e^4 - 530*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*d^4*e^5 - 40*(sqrt(
c)*x - sqrt(c*x^2 + a))^2*a^3*c^(11/2)*d^6*e^3 - 230*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*d^3*e^6 + 400*(sq
rt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^5*d^5*e^4 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*d^2*e^7 + 1170*(
sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/2)*d^4*e^5 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*d*e^8 + 910*(s
qrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*d^3*e^6 + 20*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c^5*d^5*e^4 + 570*(sqrt(c
)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*d^2*e^7 - 230*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(9/2)*d^4*e^5 + 150*(
sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*d*e^8 - 770*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^4*d^3*e^6 + 40*(sqrt(
c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*e^9 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2)*d^2*e^7 - 2*a^5*c^
(9/2)*d^4*e^5 - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*d*e^8 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^4*d
^3*e^6 + 350*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2)*d^2*e^7 + 170*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^3
*d*e^8 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(5/2)*e^9 - 9*a^6*c^(7/2)*d^2*e^7 - 65*(sqrt(c)*x - sqrt(c*x
^2 + a))*a^6*c^3*d*e^8 + 8*a^7*c^(5/2)*e^9)/((c^3*d^6*e^4 + 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 + a^3*e^10)*((sq
rt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^5)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(3/2)/(d + e*x)^6,x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^6, x)

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