3.6.84 \(\int \frac {1}{(d+e x)^3 (a+c x^2)^{5/2}} \, dx\) [584]

Optimal. Leaf size=327 \[ \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{9/2}} \]

[Out]

1/3*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)^2/(c*x^2+a)^(3/2)-5/2*c*e^4*(-a*e^2+6*c*d^2)*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)^(9/2)+1/3*(-a*e*(-5*a*e^2+2*c*d^2)+c*d*(9*a*e^2+2*c*d^2)*x)/a^2
/(a*e^2+c*d^2)^2/(e*x+d)^2/(c*x^2+a)^(1/2)+1/6*e*(-15*a^2*e^4+24*a*c*d^2*e^2+4*c^2*d^4)*(c*x^2+a)^(1/2)/a^2/(a
*e^2+c*d^2)^3/(e*x+d)^2+1/6*c*d*e*(-81*a^2*e^4+28*a*c*d^2*e^2+4*c^2*d^4)*(c*x^2+a)^(1/2)/a^2/(a*e^2+c*d^2)^4/(
e*x+d)

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Rubi [A]
time = 0.24, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {755, 837, 849, 821, 739, 212} \begin {gather*} \frac {c d e \sqrt {a+c x^2} \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x) \left (a e^2+c d^2\right )^4}+\frac {e \sqrt {a+c x^2} \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d x \left (9 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {5 c e^4 \left (6 c d^2-a e^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 \left (a e^2+c d^2\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)^2*(a + c*x^2)^(3/2)) - (a*e*(2*c*d^2 - 5*a*e^2) - c*d*(2*c*d^2 +
9*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)^2*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^
4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*d*e*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*S
qrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^4*(d + e*x)) - (5*c*e^4*(6*c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c
*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(9/2))

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 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 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(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
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 849

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{5/2}} \, dx &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 c d^2-5 a e^2-4 c d e x}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx}{3 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {\int \frac {-3 a c e^2 \left (2 c d^2-5 a e^2\right )+2 c^2 d e \left (2 c d^2+9 a e^2\right ) x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {\int \frac {2 a c^2 d e^2 \left (2 c d^2-33 a e^2\right )-c^2 e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{6 a^2 c \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {\left (5 c e^4 \left (6 c d^2-a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {\left (5 c e^4 \left (6 c d^2-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 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {5 c e^4 \left (6 c d^2-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 \left (c d^2+a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 10.66, size = 296, normalized size = 0.91 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {a+c x^2} \left (-\frac {3 e^5 \left (c d^2+a e^2\right )}{(d+e x)^2}-\frac {33 c d e^5}{d+e x}+\frac {4 c \left (-3 a^3 e^5+c^3 d^5 x+7 a c^2 d^3 e^2 x+3 a^2 c d e^3 (5 d-4 e x)\right )}{a^2 \left (a+c x^2\right )}+\frac {2 c \left (c d^2+a e^2\right ) \left (-a^2 e^3+c^2 d^3 x+3 a c d e (d-e x)\right )}{a \left (a+c x^2\right )^2}\right )}{\left (c d^2+a e^2\right )^4}+\frac {15 c e^4 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{9/2}}+\frac {15 c e^4 \left (-6 c d^2+a e^2\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{9/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]

[Out]

((Sqrt[a + c*x^2]*((-3*e^5*(c*d^2 + a*e^2))/(d + e*x)^2 - (33*c*d*e^5)/(d + e*x) + (4*c*(-3*a^3*e^5 + c^3*d^5*
x + 7*a*c^2*d^3*e^2*x + 3*a^2*c*d*e^3*(5*d - 4*e*x)))/(a^2*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)*(-(a^2*e^3) + c
^2*d^3*x + 3*a*c*d*e*(d - e*x)))/(a*(a + c*x^2)^2)))/(c*d^2 + a*e^2)^4 + (15*c*e^4*(6*c*d^2 - a*e^2)*Log[d + e
*x])/(c*d^2 + a*e^2)^(9/2) + (15*c*e^4*(-6*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2
]])/(c*d^2 + a*e^2)^(9/2))/6

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1602\) vs. \(2(303)=606\).
time = 0.51, size = 1603, normalized size = 4.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1603\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e^3*(-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)^(3/2)+7/2*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)^(3/2)+5*c*d*e/(a*e^2+c*d
^2)*(1/3/(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)^(3/2)+c*d*e/(a*e^2+c*d^2)*(2/3*(2*c
*(x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+
16/3*c/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)^2*(2*c*(x+d/e)-2*c*d/e)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2
)/e^2)^(1/2))+1/(a*e^2+c*d^2)*e^2*(1/(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)+2
*c*d*e/(a*e^2+c*d^2)*(2*c*(x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*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)-1/(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*(2/3*(2*c*(x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*
d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)^2*(2*c*(x+d/e)-2*c*d/e)/(c*(x+d/e)^2-2*c*d/e*(x+d
/e)+(a*e^2+c*d^2)/e^2)^(1/2)))-5/2*c/(a*e^2+c*d^2)*e^2*(1/3/(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)^(3/2)+c*d*e/(a*e^2+c*d^2)*(2/3*(2*c*(x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/(c*
(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)^2*(2*c*(x+d/e)
-2*c*d/e)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))+1/(a*e^2+c*d^2)*e^2*(1/(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)+2*c*d*e/(a*e^2+c*d^2)*(2*c*(x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^
2)/e^2-4*c^2*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)-1/(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)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1149 vs. \(2 (299) = 598\).
time = 0.37, size = 1149, normalized size = 3.51 \begin {gather*} \frac {35 \, c^{3} d^{3} x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{4} d^{8} e^{\left (-2\right )} + 4 \, \sqrt {c x^{2} + a} a^{2} c^{3} d^{6} + 6 \, \sqrt {c x^{2} + a} a^{3} c^{2} d^{4} e^{2} + 4 \, \sqrt {c x^{2} + a} a^{4} c d^{2} e^{4} + \sqrt {c x^{2} + a} a^{5} e^{6}\right )}} + \frac {35 \, c^{3} d^{3} x}{6 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c^{3} d^{6} + 3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} c^{2} d^{4} e^{2} + 3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} c d^{2} e^{4} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{4} e^{6}\right )}} + \frac {35 \, c^{3} d^{3} x}{3 \, {\left (\sqrt {c x^{2} + a} a^{2} c^{3} d^{6} + 3 \, \sqrt {c x^{2} + a} a^{3} c^{2} d^{4} e^{2} + 3 \, \sqrt {c x^{2} + a} a^{4} c d^{2} e^{4} + \sqrt {c x^{2} + a} a^{5} e^{6}\right )}} + \frac {35 \, c^{2} d^{2}}{2 \, {\left (\sqrt {c x^{2} + a} c^{4} d^{8} e^{\left (-3\right )} + 4 \, \sqrt {c x^{2} + a} a c^{3} d^{6} e^{\left (-1\right )} + 6 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} e + 4 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{3} + \sqrt {c x^{2} + a} a^{4} e^{5}\right )}} + \frac {35 \, c^{2} d^{2}}{6 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3} d^{6} e^{\left (-1\right )} + 3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a c^{2} d^{4} e + 3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d^{2} e^{3} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} e^{5}\right )}} - \frac {5 \, c^{2} d x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{3} d^{6} e^{\left (-2\right )} + 3 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} + 3 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{4} e^{4}\right )}} - \frac {11 \, c^{2} d x}{2 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c^{2} d^{4} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d^{2} e^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} e^{4}\right )}} - \frac {11 \, c^{2} d x}{\sqrt {c x^{2} + a} a^{2} c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{4} e^{4}} + \frac {35 \, c^{2} d^{2} \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 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {9}{2}}} - \frac {7 \, c d}{2 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{5} e^{\left (-1\right )} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{4} x + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} x e^{2} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{3} e + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} x e^{4} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} d e^{3}\right )}} - \frac {5 \, c}{2 \, {\left (\sqrt {c x^{2} + a} c^{3} d^{6} e^{\left (-3\right )} + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{4} e^{\left (-1\right )} + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e + \sqrt {c x^{2} + a} a^{3} e^{3}\right )}} - \frac {5 \, c}{6 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{4} e^{\left (-1\right )} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} e + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{3}\right )}} - \frac {5 \, c \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 (-3\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {7}{2}}} - \frac {1}{2 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{2} x^{2} e + {\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{4} e^{\left (-1\right )} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{3} x + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a x^{2} e^{3} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d x e^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{2} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

35/2*c^3*d^3*x/(sqrt(c*x^2 + a)*a*c^4*d^8*e^(-2) + 4*sqrt(c*x^2 + a)*a^2*c^3*d^6 + 6*sqrt(c*x^2 + a)*a^3*c^2*d
^4*e^2 + 4*sqrt(c*x^2 + a)*a^4*c*d^2*e^4 + sqrt(c*x^2 + a)*a^5*e^6) + 35/6*c^3*d^3*x/((c*x^2 + a)^(3/2)*a*c^3*
d^6 + 3*(c*x^2 + a)^(3/2)*a^2*c^2*d^4*e^2 + 3*(c*x^2 + a)^(3/2)*a^3*c*d^2*e^4 + (c*x^2 + a)^(3/2)*a^4*e^6) + 3
5/3*c^3*d^3*x/(sqrt(c*x^2 + a)*a^2*c^3*d^6 + 3*sqrt(c*x^2 + a)*a^3*c^2*d^4*e^2 + 3*sqrt(c*x^2 + a)*a^4*c*d^2*e
^4 + sqrt(c*x^2 + a)*a^5*e^6) + 35/2*c^2*d^2/(sqrt(c*x^2 + a)*c^4*d^8*e^(-3) + 4*sqrt(c*x^2 + a)*a*c^3*d^6*e^(
-1) + 6*sqrt(c*x^2 + a)*a^2*c^2*d^4*e + 4*sqrt(c*x^2 + a)*a^3*c*d^2*e^3 + sqrt(c*x^2 + a)*a^4*e^5) + 35/6*c^2*
d^2/((c*x^2 + a)^(3/2)*c^3*d^6*e^(-1) + 3*(c*x^2 + a)^(3/2)*a*c^2*d^4*e + 3*(c*x^2 + a)^(3/2)*a^2*c*d^2*e^3 +
(c*x^2 + a)^(3/2)*a^3*e^5) - 5/2*c^2*d*x/(sqrt(c*x^2 + a)*a*c^3*d^6*e^(-2) + 3*sqrt(c*x^2 + a)*a^2*c^2*d^4 + 3
*sqrt(c*x^2 + a)*a^3*c*d^2*e^2 + sqrt(c*x^2 + a)*a^4*e^4) - 11/2*c^2*d*x/((c*x^2 + a)^(3/2)*a*c^2*d^4 + 2*(c*x
^2 + a)^(3/2)*a^2*c*d^2*e^2 + (c*x^2 + a)^(3/2)*a^3*e^4) - 11*c^2*d*x/(sqrt(c*x^2 + a)*a^2*c^2*d^4 + 2*sqrt(c*
x^2 + a)*a^3*c*d^2*e^2 + sqrt(c*x^2 + a)*a^4*e^4) + 35/2*c^2*d^2*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/
(sqrt(a*c)*abs(x*e + d)))*e^(-5)/(c*d^2*e^(-2) + a)^(9/2) - 7/2*c*d/((c*x^2 + a)^(3/2)*c^2*d^5*e^(-1) + (c*x^2
+ a)^(3/2)*c^2*d^4*x + 2*(c*x^2 + a)^(3/2)*a*c*d^2*x*e^2 + 2*(c*x^2 + a)^(3/2)*a*c*d^3*e + (c*x^2 + a)^(3/2)*
a^2*x*e^4 + (c*x^2 + a)^(3/2)*a^2*d*e^3) - 5/2*c/(sqrt(c*x^2 + a)*c^3*d^6*e^(-3) + 3*sqrt(c*x^2 + a)*a*c^2*d^4
*e^(-1) + 3*sqrt(c*x^2 + a)*a^2*c*d^2*e + sqrt(c*x^2 + a)*a^3*e^3) - 5/6*c/((c*x^2 + a)^(3/2)*c^2*d^4*e^(-1) +
2*(c*x^2 + a)^(3/2)*a*c*d^2*e + (c*x^2 + a)^(3/2)*a^2*e^3) - 5/2*c*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a
*e/(sqrt(a*c)*abs(x*e + d)))*e^(-3)/(c*d^2*e^(-2) + a)^(7/2) - 1/2/((c*x^2 + a)^(3/2)*c*d^2*x^2*e + (c*x^2 + a
)^(3/2)*c*d^4*e^(-1) + 2*(c*x^2 + a)^(3/2)*c*d^3*x + (c*x^2 + a)^(3/2)*a*x^2*e^3 + 2*(c*x^2 + a)^(3/2)*a*d*x*e
^2 + (c*x^2 + a)^(3/2)*a*d^2*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (299) = 598\).
time = 9.18, size = 2590, normalized size = 7.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(15*sqrt(c*d^2 + a*e^2)*((a^3*c^3*x^6 + 2*a^4*c^2*x^4 + a^5*c*x^2)*e^8 + 2*(a^3*c^3*d*x^5 + 2*a^4*c^2*d
*x^3 + a^5*c*d*x)*e^7 - (6*a^2*c^4*d^2*x^6 + 11*a^3*c^3*d^2*x^4 + 4*a^4*c^2*d^2*x^2 - a^5*c*d^2)*e^6 - 12*(a^2
*c^4*d^3*x^5 + 2*a^3*c^3*d^3*x^3 + a^4*c^2*d^3*x)*e^5 - 6*(a^2*c^4*d^4*x^4 + 2*a^3*c^3*d^4*x^2 + a^4*c^2*d^4)*
e^4)*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*(4*c^6*d^9*x^3 + 6*a*c^5*d^9*x - (15*a^4*c^2*x^4 + 20*a^5*c*x
^2 + 3*a^6)*e^9 - (81*a^3*c^3*d*x^5 + 144*a^4*c^2*d*x^3 + 61*a^5*c*d*x)*e^8 - (87*a^3*c^3*d^2*x^4 + 148*a^4*c^
2*d^2*x^2 + 53*a^5*c*d^2)*e^7 - (53*a^2*c^4*d^3*x^5 + 48*a^3*c^3*d^3*x^3 - 13*a^4*c^2*d^3*x)*e^6 - 2*(8*a^2*c^
4*d^4*x^4 + 7*a^3*c^3*d^4*x^2 - 7*a^4*c^2*d^4)*e^5 + 2*(16*a*c^5*d^5*x^5 + 65*a^2*c^4*d^5*x^3 + 55*a^3*c^3*d^5
*x)*e^4 + 2*(32*a*c^5*d^6*x^4 + 63*a^2*c^4*d^6*x^2 + 35*a^3*c^3*d^6)*e^3 + 2*(2*c^6*d^7*x^5 + 19*a*c^5*d^7*x^3
+ 21*a^2*c^4*d^7*x)*e^2 + 2*(4*c^6*d^8*x^4 + 6*a*c^5*d^8*x^2 + 3*a^2*c^4*d^8)*e)*sqrt(c*x^2 + a))/(a^2*c^7*d^
12*x^4 + 2*a^3*c^6*d^12*x^2 + a^4*c^5*d^12 + (a^7*c^2*x^6 + 2*a^8*c*x^4 + a^9*x^2)*e^12 + 2*(a^7*c^2*d*x^5 + 2
*a^8*c*d*x^3 + a^9*d*x)*e^11 + (5*a^6*c^3*d^2*x^6 + 11*a^7*c^2*d^2*x^4 + 7*a^8*c*d^2*x^2 + a^9*d^2)*e^10 + 10*
(a^6*c^3*d^3*x^5 + 2*a^7*c^2*d^3*x^3 + a^8*c*d^3*x)*e^9 + 5*(2*a^5*c^4*d^4*x^6 + 5*a^6*c^3*d^4*x^4 + 4*a^7*c^2
*d^4*x^2 + a^8*c*d^4)*e^8 + 20*(a^5*c^4*d^5*x^5 + 2*a^6*c^3*d^5*x^3 + a^7*c^2*d^5*x)*e^7 + 10*(a^4*c^5*d^6*x^6
+ 3*a^5*c^4*d^6*x^4 + 3*a^6*c^3*d^6*x^2 + a^7*c^2*d^6)*e^6 + 20*(a^4*c^5*d^7*x^5 + 2*a^5*c^4*d^7*x^3 + a^6*c^
3*d^7*x)*e^5 + 5*(a^3*c^6*d^8*x^6 + 4*a^4*c^5*d^8*x^4 + 5*a^5*c^4*d^8*x^2 + 2*a^6*c^3*d^8)*e^4 + 10*(a^3*c^6*d
^9*x^5 + 2*a^4*c^5*d^9*x^3 + a^5*c^4*d^9*x)*e^3 + (a^2*c^7*d^10*x^6 + 7*a^3*c^6*d^10*x^4 + 11*a^4*c^5*d^10*x^2
+ 5*a^5*c^4*d^10)*e^2 + 2*(a^2*c^7*d^11*x^5 + 2*a^3*c^6*d^11*x^3 + a^4*c^5*d^11*x)*e), -1/6*(15*sqrt(-c*d^2 -
a*e^2)*((a^3*c^3*x^6 + 2*a^4*c^2*x^4 + a^5*c*x^2)*e^8 + 2*(a^3*c^3*d*x^5 + 2*a^4*c^2*d*x^3 + a^5*c*d*x)*e^7 -
(6*a^2*c^4*d^2*x^6 + 11*a^3*c^3*d^2*x^4 + 4*a^4*c^2*d^2*x^2 - a^5*c*d^2)*e^6 - 12*(a^2*c^4*d^3*x^5 + 2*a^3*c^
3*d^3*x^3 + a^4*c^2*d^3*x)*e^5 - 6*(a^2*c^4*d^4*x^4 + 2*a^3*c^3*d^4*x^2 + a^4*c^2*d^4)*e^4)*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)) - (4*c^6*d^9*x^3 + 6*a
*c^5*d^9*x - (15*a^4*c^2*x^4 + 20*a^5*c*x^2 + 3*a^6)*e^9 - (81*a^3*c^3*d*x^5 + 144*a^4*c^2*d*x^3 + 61*a^5*c*d*
x)*e^8 - (87*a^3*c^3*d^2*x^4 + 148*a^4*c^2*d^2*x^2 + 53*a^5*c*d^2)*e^7 - (53*a^2*c^4*d^3*x^5 + 48*a^3*c^3*d^3*
x^3 - 13*a^4*c^2*d^3*x)*e^6 - 2*(8*a^2*c^4*d^4*x^4 + 7*a^3*c^3*d^4*x^2 - 7*a^4*c^2*d^4)*e^5 + 2*(16*a*c^5*d^5*
x^5 + 65*a^2*c^4*d^5*x^3 + 55*a^3*c^3*d^5*x)*e^4 + 2*(32*a*c^5*d^6*x^4 + 63*a^2*c^4*d^6*x^2 + 35*a^3*c^3*d^6)*
e^3 + 2*(2*c^6*d^7*x^5 + 19*a*c^5*d^7*x^3 + 21*a^2*c^4*d^7*x)*e^2 + 2*(4*c^6*d^8*x^4 + 6*a*c^5*d^8*x^2 + 3*a^2
*c^4*d^8)*e)*sqrt(c*x^2 + a))/(a^2*c^7*d^12*x^4 + 2*a^3*c^6*d^12*x^2 + a^4*c^5*d^12 + (a^7*c^2*x^6 + 2*a^8*c*x
^4 + a^9*x^2)*e^12 + 2*(a^7*c^2*d*x^5 + 2*a^8*c*d*x^3 + a^9*d*x)*e^11 + (5*a^6*c^3*d^2*x^6 + 11*a^7*c^2*d^2*x^
4 + 7*a^8*c*d^2*x^2 + a^9*d^2)*e^10 + 10*(a^6*c^3*d^3*x^5 + 2*a^7*c^2*d^3*x^3 + a^8*c*d^3*x)*e^9 + 5*(2*a^5*c^
4*d^4*x^6 + 5*a^6*c^3*d^4*x^4 + 4*a^7*c^2*d^4*x^2 + a^8*c*d^4)*e^8 + 20*(a^5*c^4*d^5*x^5 + 2*a^6*c^3*d^5*x^3 +
a^7*c^2*d^5*x)*e^7 + 10*(a^4*c^5*d^6*x^6 + 3*a^5*c^4*d^6*x^4 + 3*a^6*c^3*d^6*x^2 + a^7*c^2*d^6)*e^6 + 20*(a^4
*c^5*d^7*x^5 + 2*a^5*c^4*d^7*x^3 + a^6*c^3*d^7*x)*e^5 + 5*(a^3*c^6*d^8*x^6 + 4*a^4*c^5*d^8*x^4 + 5*a^5*c^4*d^8
*x^2 + 2*a^6*c^3*d^8)*e^4 + 10*(a^3*c^6*d^9*x^5 + 2*a^4*c^5*d^9*x^3 + a^5*c^4*d^9*x)*e^3 + (a^2*c^7*d^10*x^6 +
7*a^3*c^6*d^10*x^4 + 11*a^4*c^5*d^10*x^2 + 5*a^5*c^4*d^10)*e^2 + 2*(a^2*c^7*d^11*x^5 + 2*a^3*c^6*d^11*x^3 + a
^4*c^5*d^11*x)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(d + e*x)**3), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2044 vs. \(2 (299) = 598\).
time = 3.90, size = 2044, normalized size = 6.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

5*(6*c^2*d^2*e^4 - a*c*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^4*
d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d^2 - a*e^2)) + 1/3*((2*((c^18*
d^29 + 19*a*c^17*d^27*e^2 + 138*a^2*c^16*d^25*e^4 + 538*a^3*c^15*d^23*e^6 + 1243*a^4*c^14*d^21*e^8 + 1617*a^5*
c^13*d^19*e^10 + 528*a^6*c^12*d^17*e^12 - 2244*a^7*c^11*d^15*e^14 - 5049*a^8*c^10*d^13*e^16 - 5819*a^9*c^9*d^1
1*e^18 - 4334*a^10*c^8*d^9*e^20 - 2166*a^11*c^7*d^7*e^22 - 707*a^12*c^6*d^5*e^24 - 137*a^13*c^5*d^3*e^26 - 12*
a^14*c^4*d*e^28)*x/(a^2*c^17*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 182
0*a^6*c^13*d^24*e^8 + 4368*a^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^1
0*c^9*d^16*e^16 + 11440*a^11*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5
*d^8*e^24 + 560*a^15*c^4*d^6*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32) + 3*(5*a^2*c^1
6*d^26*e^3 + 59*a^3*c^15*d^24*e^5 + 318*a^4*c^14*d^22*e^7 + 1034*a^5*c^13*d^20*e^9 + 2255*a^6*c^12*d^18*e^11 +
3465*a^7*c^11*d^16*e^13 + 3828*a^8*c^10*d^14*e^15 + 3036*a^9*c^9*d^12*e^17 + 1683*a^10*c^8*d^10*e^19 + 605*a^
11*c^7*d^8*e^21 + 110*a^12*c^6*d^6*e^23 - 6*a^13*c^5*d^4*e^25 - 7*a^14*c^4*d^2*e^27 - a^15*c^3*e^29)/(a^2*c^17
*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 4368*a
^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*a^11
*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4*d^6
*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))*x + 3*(a*c^17*d^29 + 16*a^2*c^16*d^27*e^2
+ 105*a^3*c^15*d^25*e^4 + 376*a^4*c^14*d^23*e^6 + 781*a^5*c^13*d^21*e^8 + 792*a^6*c^12*d^19*e^10 - 363*a^7*c^
11*d^17*e^12 - 2640*a^8*c^10*d^15*e^14 - 4653*a^9*c^9*d^13*e^16 - 4928*a^10*c^8*d^11*e^18 - 3509*a^11*c^7*d^9*
e^20 - 1704*a^12*c^6*d^7*e^22 - 545*a^13*c^5*d^5*e^24 - 104*a^14*c^4*d^3*e^26 - 9*a^15*c^3*d*e^28)/(a^2*c^17*d
^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 4368*a^7
*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*a^11*c
^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4*d^6*e
^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))*x + (3*a^2*c^16*d^28*e + 68*a^3*c^15*d^26*e
^3 + 575*a^4*c^14*d^24*e^5 + 2688*a^5*c^13*d^22*e^7 + 8063*a^6*c^12*d^20*e^9 + 16676*a^7*c^11*d^18*e^11 + 2465
1*a^8*c^10*d^16*e^13 + 26400*a^9*c^9*d^14*e^15 + 20361*a^10*c^8*d^12*e^17 + 10956*a^11*c^7*d^10*e^19 + 3773*a^
12*c^6*d^8*e^21 + 608*a^13*c^5*d^6*e^23 - 75*a^14*c^4*d^4*e^25 - 52*a^15*c^3*d^2*e^27 - 7*a^16*c^2*e^29)/(a^2*
c^17*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 43
68*a^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*
a^11*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4
*d^6*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))/(c*x^2 + a)^(3/2) - (22*(sqrt(c)*x -
sqrt(c*x^2 + a))^2*c^(5/2)*d^3*e^4 + 10*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*d^2*e^5 - 34*(sqrt(c)*x - sqrt(c*x
^2 + a))*a*c^2*d^2*e^5 - 11*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*e^6 - (sqrt(c)*x - sqrt(c*x^2 + a))^3*
a*c*e^7 + 11*a^2*c^(3/2)*d*e^6 - (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*e^7)/((c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*
c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*
sqrt(c)*d - a*e)^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)^(5/2)*(d + e*x)^3),x)

[Out]

int(1/((a + c*x^2)^(5/2)*(d + e*x)^3), x)

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