3.6.41 \(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^4} \, dx\) [541]

Optimal. Leaf size=200 \[ -\frac {c \left (d \left (2 c d^2+a e^2\right )+e \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{2 e^3 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}+\frac {c^2 d \left (2 c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 e^4 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

-1/3*(c*x^2+a)^(3/2)/e/(e*x+d)^3+c^(3/2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/e^4+1/2*c^2*d*(3*a*e^2+2*c*d^2)*ar
ctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^4/(a*e^2+c*d^2)^(3/2)-1/2*c*(d*(a*e^2+2*c*d^2)+e*(2*
a*e^2+3*c*d^2)*x)*(c*x^2+a)^(1/2)/e^3/(a*e^2+c*d^2)/(e*x+d)^2

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Rubi [A]
time = 0.26, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {747, 825, 858, 223, 212, 739} \begin {gather*} \frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}+\frac {c^2 d \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac {c \sqrt {a+c x^2} \left (e x \left (2 a e^2+3 c d^2\right )+d \left (a e^2+2 c d^2\right )\right )}{2 e^3 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

-1/2*(c*(d*(2*c*d^2 + a*e^2) + e*(3*c*d^2 + 2*a*e^2)*x)*Sqrt[a + c*x^2])/(e^3*(c*d^2 + a*e^2)*(d + e*x)^2) - (
a + c*x^2)^(3/2)/(3*e*(d + e*x)^3) + (c^(3/2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^4 + (c^2*d*(2*c*d^2 + 3*
a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*e^4*(c*d^2 + a*e^2)^(3/2))

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 747

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

Rule 825

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

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 1.79, size = 210, normalized size = 1.05 \begin {gather*} -\frac {\frac {e \sqrt {a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+9 d e x+8 e^2 x^2\right )+c^2 d^2 \left (6 d^2+15 d e x+11 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right ) (d+e x)^3}+\frac {6 c^2 d \left (2 c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+6 c^{3/2} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

-1/6*((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 9*d*e*x + 8*e^2*x^2) + c^2*d^2*(6*d^2 + 15*d*e*x + 11*e
^2*x^2)))/((c*d^2 + a*e^2)*(d + e*x)^3) + (6*c^2*d*(2*c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a +
c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2) - a*e^2)^(3/2) + 6*c^(3/2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/e^4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2445\) vs. \(2(178)=356\).
time = 0.44, size = 2446, normalized size = 12.23

method	result	size
default	\(\text {Expression too large to display}\)	\(2446\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/e^4*(-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))))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (173) = 346\).
time = 0.33, size = 619, normalized size = 3.10 \begin {gather*} \frac {c^{3} 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 (-7\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + a} c^{3} d^{3}}{2 \, {\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )}} - \frac {\sqrt {c x^{2} + a} c^{3} d^{2} x}{2 \, {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{2}}{6 \, {\left (c^{2} d^{4} x e^{2} + c^{2} d^{5} e + 2 \, a c d^{2} x e^{4} + 2 \, a c d^{3} e^{3} + a^{2} x e^{6} + a^{2} d e^{5}\right )}} - \frac {3 \, c^{2} 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 (-5\right )}}{2 \, \sqrt {c d^{2} e^{\left (-2\right )} + a}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c d}{6 \, {\left (c^{2} d^{4} x^{2} e + c^{2} d^{6} e^{\left (-1\right )} + 2 \, c^{2} d^{5} x + 2 \, a c d^{2} x^{2} e^{3} + 4 \, a c d^{3} x e^{2} + 2 \, a c d^{4} e + a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4} + a^{2} d^{2} e^{3}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d}{6 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + c^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )} - \frac {3 \, \sqrt {c x^{2} + a} c^{2} d}{2 \, {\left (c d^{2} e^{3} + a e^{5}\right )}} + \frac {\sqrt {c x^{2} + a} c^{2} x}{c d^{2} e^{2} + a e^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{3 \, {\left (c d^{2} x^{3} e^{2} + 3 \, c d^{3} x^{2} e + c d^{5} e^{\left (-1\right )} + 3 \, c d^{4} x + a x^{3} e^{4} + 3 \, a d x^{2} e^{3} + 3 \, a d^{2} x e^{2} + a d^{3} e\right )}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c}{3 \, {\left (c d^{2} x e^{2} + c d^{3} e + a x e^{4} + a d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/2*c^3*d^3*arcsinh(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) + 1/2*sqrt(c*x^2 + a)*c^3*d^3/(c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2*e^7) - 1/2*sqrt(c*x^2 + a)*c^3*d^2*x/(c^
2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) + 1/6*(c*x^2 + a)^(3/2)*c^2*d^2/(c^2*d^4*x*e^2 + c^2*d^5*e + 2*a*c*d^2*x*
e^4 + 2*a*c*d^3*e^3 + a^2*x*e^6 + a^2*d*e^5) - 3/2*c^2*d*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*
c)*abs(x*e + d)))*e^(-5)/sqrt(c*d^2*e^(-2) + a) - 1/6*(c*x^2 + a)^(5/2)*c*d/(c^2*d^4*x^2*e + c^2*d^6*e^(-1) +
2*c^2*d^5*x + 2*a*c*d^2*x^2*e^3 + 4*a*c*d^3*x*e^2 + 2*a*c*d^4*e + a^2*x^2*e^5 + 2*a^2*d*x*e^4 + a^2*d^2*e^3) +
1/6*(c*x^2 + a)^(3/2)*c^2*d/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) + c^(3/2)*arcsinh(c*x/sqrt(a*c))*e^(-4) - 3
/2*sqrt(c*x^2 + a)*c^2*d/(c*d^2*e^3 + a*e^5) + sqrt(c*x^2 + a)*c^2*x/(c*d^2*e^2 + a*e^4) - 1/3*(c*x^2 + a)^(5/
2)/(c*d^2*x^3*e^2 + 3*c*d^3*x^2*e + c*d^5*e^(-1) + 3*c*d^4*x + a*x^3*e^4 + 3*a*d*x^2*e^3 + 3*a*d^2*x*e^2 + a*d
^3*e) - 2/3*(c*x^2 + a)^(3/2)*c/(c*d^2*x*e^2 + c*d^3*e + a*x*e^4 + a*d*e^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (173) = 346\).
time = 8.62, size = 2447, normalized size = 12.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/12*(6*(3*c^3*d^6*x*e + c^3*d^7 + a^2*c*x^3*e^7 + 3*a^2*c*d*x^2*e^6 + (2*a*c^2*d^2*x^3 + 3*a^2*c*d^2*x)*e^5
+ (6*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + (c^3*d^4*x^3 + 6*a*c^2*d^4*x)*e^3 + (3*c^3*d^5*x^2 + 2*a*c^2*d^5)*e^2)*s
qrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*(6*c^3*d^5*x*e + 2*c^3*d^6 + 3*a*c^2*d*x^3*e^5 + 9*
a*c^2*d^2*x^2*e^4 + (2*c^3*d^3*x^3 + 9*a*c^2*d^3*x)*e^3 + 3*(2*c^3*d^4*x^2 + a*c^2*d^4)*e^2)*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*(15*c^3*d^5*x*e^2 + 6*c^3*d^6*e + 24*a*c^2*d^3*x*e^4 + 9*a^2*c*
d*x*e^6 + 2*(4*a^2*c*x^2 + a^3)*e^7 + (19*a*c^2*d^2*x^2 + 7*a^2*c*d^2)*e^5 + 11*(c^3*d^4*x^2 + a*c^2*d^4)*e^3)
*sqrt(c*x^2 + a))/(3*c^2*d^6*x*e^5 + c^2*d^7*e^4 + a^2*x^3*e^11 + 3*a^2*d*x^2*e^10 + (2*a*c*d^2*x^3 + 3*a^2*d^
2*x)*e^9 + (6*a*c*d^3*x^2 + a^2*d^3)*e^8 + (c^2*d^4*x^3 + 6*a*c*d^4*x)*e^7 + (3*c^2*d^5*x^2 + 2*a*c*d^5)*e^6),
-1/6*(3*(6*c^3*d^5*x*e + 2*c^3*d^6 + 3*a*c^2*d*x^3*e^5 + 9*a*c^2*d^2*x^2*e^4 + (2*c^3*d^3*x^3 + 9*a*c^2*d^3*x
)*e^3 + 3*(2*c^3*d^4*x^2 + a*c^2*d^4)*e^2)*sqrt(-c*d^2 - a*e^2)*arctan(-sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqr
t(c*x^2 + a)/(c^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) - 3*(3*c^3*d^6*x*e + c^3*d^7 + a^2*c*x^3*e^7 + 3*a
^2*c*d*x^2*e^6 + (2*a*c^2*d^2*x^3 + 3*a^2*c*d^2*x)*e^5 + (6*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + (c^3*d^4*x^3 + 6*
a*c^2*d^4*x)*e^3 + (3*c^3*d^5*x^2 + 2*a*c^2*d^5)*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a)
+ (15*c^3*d^5*x*e^2 + 6*c^3*d^6*e + 24*a*c^2*d^3*x*e^4 + 9*a^2*c*d*x*e^6 + 2*(4*a^2*c*x^2 + a^3)*e^7 + (19*a*c
^2*d^2*x^2 + 7*a^2*c*d^2)*e^5 + 11*(c^3*d^4*x^2 + a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c^2*d^6*x*e^5 + c^2*d^7*
e^4 + a^2*x^3*e^11 + 3*a^2*d*x^2*e^10 + (2*a*c*d^2*x^3 + 3*a^2*d^2*x)*e^9 + (6*a*c*d^3*x^2 + a^2*d^3)*e^8 + (c
^2*d^4*x^3 + 6*a*c*d^4*x)*e^7 + (3*c^2*d^5*x^2 + 2*a*c*d^5)*e^6), -1/12*(12*(3*c^3*d^6*x*e + c^3*d^7 + a^2*c*x
^3*e^7 + 3*a^2*c*d*x^2*e^6 + (2*a*c^2*d^2*x^3 + 3*a^2*c*d^2*x)*e^5 + (6*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + (c^3*
d^4*x^3 + 6*a*c^2*d^4*x)*e^3 + (3*c^3*d^5*x^2 + 2*a*c^2*d^5)*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a))
- 3*(6*c^3*d^5*x*e + 2*c^3*d^6 + 3*a*c^2*d*x^3*e^5 + 9*a*c^2*d^2*x^2*e^4 + (2*c^3*d^3*x^3 + 9*a*c^2*d^3*x)*e^3
+ 3*(2*c^3*d^4*x^2 + a*c^2*d^4)*e^2)*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*(15*c^3*
d^5*x*e^2 + 6*c^3*d^6*e + 24*a*c^2*d^3*x*e^4 + 9*a^2*c*d*x*e^6 + 2*(4*a^2*c*x^2 + a^3)*e^7 + (19*a*c^2*d^2*x^2
+ 7*a^2*c*d^2)*e^5 + 11*(c^3*d^4*x^2 + a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c^2*d^6*x*e^5 + c^2*d^7*e^4 + a^2*
x^3*e^11 + 3*a^2*d*x^2*e^10 + (2*a*c*d^2*x^3 + 3*a^2*d^2*x)*e^9 + (6*a*c*d^3*x^2 + a^2*d^3)*e^8 + (c^2*d^4*x^3
+ 6*a*c*d^4*x)*e^7 + (3*c^2*d^5*x^2 + 2*a*c*d^5)*e^6), -1/6*(3*(6*c^3*d^5*x*e + 2*c^3*d^6 + 3*a*c^2*d*x^3*e^5
+ 9*a*c^2*d^2*x^2*e^4 + (2*c^3*d^3*x^3 + 9*a*c^2*d^3*x)*e^3 + 3*(2*c^3*d^4*x^2 + a*c^2*d^4)*e^2)*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)) + 6*(3*c^3*d^6*x*e + c^3*d^7 + a^2*c*x^3*e^7 + 3*a^2*c*d*x^2*e^6 + (2*a*c^2*d^2*x^3 + 3*a^2*c*d^2*x)*e^5
+ (6*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + (c^3*d^4*x^3 + 6*a*c^2*d^4*x)*e^3 + (3*c^3*d^5*x^2 + 2*a*c^2*d^5)*e^2)*s
qrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (15*c^3*d^5*x*e^2 + 6*c^3*d^6*e + 24*a*c^2*d^3*x*e^4 + 9*a^2*c*d*
x*e^6 + 2*(4*a^2*c*x^2 + a^3)*e^7 + (19*a*c^2*d^2*x^2 + 7*a^2*c*d^2)*e^5 + 11*(c^3*d^4*x^2 + a*c^2*d^4)*e^3)*s
qrt(c*x^2 + a))/(3*c^2*d^6*x*e^5 + c^2*d^7*e^4 + a^2*x^3*e^11 + 3*a^2*d*x^2*e^10 + (2*a*c*d^2*x^3 + 3*a^2*d^2*
x)*e^9 + (6*a*c*d^3*x^2 + a^2*d^3)*e^8 + (c^2*d^4*x^3 + 6*a*c*d^4*x)*e^7 + (3*c^2*d^5*x^2 + 2*a*c*d^5)*e^6)]

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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 )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**4,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (173) = 346\).
time = 0.87, size = 589, normalized size = 2.94 \begin {gather*} -c^{\frac {3}{2}} e^{\left (-4\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \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 )}{{\left (c d^{2} e^{4} + a e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {54 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{2} - 78 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e - 34 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{3} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{4} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} d^{2} e^{3} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{5} - 11 \, a^{3} c^{\frac {5}{2}} d^{2} e^{3} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{4} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} e^{5} - 8 \, a^{4} c^{\frac {3}{2}} e^{5}}{3 \, {\left (c d^{2} e^{4} + a e^{6}\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 )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-c^(3/2)*e^(-4)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + (2*c^3*d^3 + 3*a*c^2*d*e^2)*arctan(((sqrt(c)*x - sqrt
(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c*d^2*e^4 + a*e^6)*sqrt(-c*d^2 - a*e^2)) - 1/3*(54*(sqrt(c
)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d^5 + 18*(sqrt(c)*x - sqrt(c*x
^2 + a))^5*c^3*d^3*e^2 - 78*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e - 34*(sqrt(c)*x - sqrt(c*x^2 + a))
^3*a*c^3*d^3*e^2 + 27*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^3 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a
*c^2*d*e^4 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*d^3*e^2 - 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)
*d^2*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*e^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*e
^5 - 11*a^3*c^(5/2)*d^2*e^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^4 + 12*(sqrt(c)*x - sqrt(c*x^2 + a)
)^2*a^3*c^(3/2)*e^5 - 8*a^4*c^(3/2)*e^5)/((c*d^2*e^4 + a*e^6)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*
x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,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)^4,x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^4, x)

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