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

Optimal result

Integrand size = 19, antiderivative size = 195 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {3 a^2 c^3 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}} \]

[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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {745, 735, 739, 212} \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {3 a^2 c^3 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac {3 a c^2 d \sqrt {a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {c d \left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 3.80 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {\sqrt {a+c x^2} \left (-8 a^4 e^5+2 c^4 d^4 x^3 (5 d+e x)-2 a^3 c e^3 \left (13 d^2+5 d e x+8 e^2 x^2\right )+a c^3 d^2 x \left (25 d^3+29 d^2 e x+45 d e^2 x^2+9 e^3 x^3\right )-a^2 c^2 e \left (33 d^4+45 d^3 e x+77 d^2 e^2 x^2+25 d e^3 x^3+8 e^4 x^4\right )\right )}{40 \left (c d^2+a e^2\right )^3 (d+e x)^5}+\frac {3 a^2 c^3 d \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{4 \left (-c d^2-a e^2\right )^{7/2}} \]

[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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4107\) vs. \(2(175)=350\).

Time = 2.13 (sec) , antiderivative size = 4108, normalized size of antiderivative = 21.07

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

[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/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)))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 810 vs. \(2 (176) = 352\).

Time = 3.34 (sec) , antiderivative size = 1647, normalized size of antiderivative = 8.45 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/80*(15*(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*x^4 + 10*a^2*c^3*d^3*e^3*x^3 + 10*a^2*c^3*d^4*e^2*x^2 + 5*a^2
*c^3*d^5*e*x + a^2*c^3*d^6)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)
*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(33*a^2*c^3*d^6*e +
59*a^3*c^2*d^4*e^3 + 34*a^4*c*d^2*e^5 + 8*a^5*e^7 - (2*c^5*d^6*e + 11*a*c^4*d^4*e^3 + a^2*c^3*d^2*e^5 - 8*a^3
*c^2*e^7)*x^4 - 5*(2*c^5*d^7 + 11*a*c^4*d^5*e^2 + 4*a^2*c^3*d^3*e^4 - 5*a^3*c^2*d*e^6)*x^3 - (29*a*c^4*d^6*e -
48*a^2*c^3*d^4*e^3 - 93*a^3*c^2*d^2*e^5 - 16*a^4*c*e^7)*x^2 - 5*(5*a*c^4*d^7 - 4*a^2*c^3*d^5*e^2 - 11*a^3*c^2
*d^3*e^4 - 2*a^4*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^13 + 4*a*c^3*d^11*e^2 + 6*a^2*c^2*d^9*e^4 + 4*a^3*c*d^7*e
^6 + a^4*d^5*e^8 + (c^4*d^8*e^5 + 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^9 + 4*a^3*c*d^2*e^11 + a^4*e^13)*x^5 + 5*(
c^4*d^9*e^4 + 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8 + 4*a^3*c*d^3*e^10 + a^4*d*e^12)*x^4 + 10*(c^4*d^10*e^3 + 4*
a*c^3*d^8*e^5 + 6*a^2*c^2*d^6*e^7 + 4*a^3*c*d^4*e^9 + a^4*d^2*e^11)*x^3 + 10*(c^4*d^11*e^2 + 4*a*c^3*d^9*e^4 +
6*a^2*c^2*d^7*e^6 + 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)*x^2 + 5*(c^4*d^12*e + 4*a*c^3*d^10*e^3 + 6*a^2*c^2*d^8*e^
5 + 4*a^3*c*d^6*e^7 + a^4*d^4*e^9)*x), -1/40*(15*(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*x^4 + 10*a^2*c^3*d^3*e
^3*x^3 + 10*a^2*c^3*d^4*e^2*x^2 + 5*a^2*c^3*d^5*e*x + 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)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) + (33*a^2*c^3*d^6*e + 59*a^
3*c^2*d^4*e^3 + 34*a^4*c*d^2*e^5 + 8*a^5*e^7 - (2*c^5*d^6*e + 11*a*c^4*d^4*e^3 + a^2*c^3*d^2*e^5 - 8*a^3*c^2*e
^7)*x^4 - 5*(2*c^5*d^7 + 11*a*c^4*d^5*e^2 + 4*a^2*c^3*d^3*e^4 - 5*a^3*c^2*d*e^6)*x^3 - (29*a*c^4*d^6*e - 48*a^
2*c^3*d^4*e^3 - 93*a^3*c^2*d^2*e^5 - 16*a^4*c*e^7)*x^2 - 5*(5*a*c^4*d^7 - 4*a^2*c^3*d^5*e^2 - 11*a^3*c^2*d^3*e
^4 - 2*a^4*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^13 + 4*a*c^3*d^11*e^2 + 6*a^2*c^2*d^9*e^4 + 4*a^3*c*d^7*e^6 + a
^4*d^5*e^8 + (c^4*d^8*e^5 + 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^9 + 4*a^3*c*d^2*e^11 + a^4*e^13)*x^5 + 5*(c^4*d^
9*e^4 + 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8 + 4*a^3*c*d^3*e^10 + a^4*d*e^12)*x^4 + 10*(c^4*d^10*e^3 + 4*a*c^3*
d^8*e^5 + 6*a^2*c^2*d^6*e^7 + 4*a^3*c*d^4*e^9 + a^4*d^2*e^11)*x^3 + 10*(c^4*d^11*e^2 + 4*a*c^3*d^9*e^4 + 6*a^2
*c^2*d^7*e^6 + 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)*x^2 + 5*(c^4*d^12*e + 4*a*c^3*d^10*e^3 + 6*a^2*c^2*d^8*e^5 + 4*
a^3*c*d^6*e^7 + a^4*d^4*e^9)*x)]

Sympy [F]

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

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (176) = 352\).

Time = 0.35 (sec) , antiderivative size = 1283, normalized size of antiderivative = 6.58 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x \]

[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)