3.6.83 \(\int \frac {1}{(d+e x)^2 (a+c x^2)^{5/2}} \, dx\) [583]

Optimal. Leaf size=244 \[ \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {5 c d e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{7/2}} \]

[Out]

1/3*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)/(c*x^2+a)^(3/2)-5*c*d*e^4*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)+1/3*(-a*e*(-4*a*e^2+c*d^2)+c*d*(7*a*e^2+2*c*d^2)*x)/a^2/(a*e^2+c*d^2)^2/(e*x
+d)/(c*x^2+a)^(1/2)+1/3*e*(-8*a^2*e^4+9*a*c*d^2*e^2+2*c^2*d^4)*(c*x^2+a)^(1/2)/a^2/(a*e^2+c*d^2)^3/(e*x+d)

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Rubi [A]
time = 0.15, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {755, 837, 821, 739, 212} \begin {gather*} \frac {e \sqrt {a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} (d+e x) \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) \left (a e^2+c d^2\right )}-\frac {5 c d e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^(3/2)) - (a*e*(c*d^2 - 4*a*e^2) - c*d*(2*c*d^2 + 7*a*
e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*Sqrt[a + c*x^2]) + (e*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)*Sqrt[
a + c*x^2])/(3*a^2*(c*d^2 + a*e^2)^3*(d + e*x)) - (5*c*d*e^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a
+ c*x^2])])/(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 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])

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^2 \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) \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 \left (c d^2+2 a e^2\right )-3 c d e x}{(d+e x)^2 \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) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {\int \frac {-2 a c e^2 \left (c d^2-4 a e^2\right )+c^2 d e \left (2 c d^2+7 a e^2\right ) x}{(d+e x)^2 \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) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {\left (5 c d e^4\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{\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) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {\left (5 c d e^4\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 )}{\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) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {5 c d e^4 \tanh ^{-1}\left (\frac {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 )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.18, size = 247, normalized size = 1.01 \begin {gather*} \frac {-3 a^4 e^5+2 c^4 d^4 x^3 (d+e x)+2 a^3 c e^3 \left (7 d^2+4 d e x-6 e^2 x^2\right )+3 a c^3 d^2 x \left (d^3+d^2 e x+3 d e^2 x^2+3 e^3 x^3\right )+a^2 c^2 e \left (2 d^4+11 d^3 e x+21 d^2 e^2 x^2+7 d e^3 x^3-8 e^4 x^4\right )}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )^{3/2}}+\frac {10 c d e^4 \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 )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^4*e^5 + 2*c^4*d^4*x^3*(d + e*x) + 2*a^3*c*e^3*(7*d^2 + 4*d*e*x - 6*e^2*x^2) + 3*a*c^3*d^2*x*(d^3 + d^2*e
*x + 3*d*e^2*x^2 + 3*e^3*x^3) + a^2*c^2*e*(2*d^4 + 11*d^3*e*x + 21*d^2*e^2*x^2 + 7*d*e^3*x^3 - 8*e^4*x^4))/(3*
a^2*(c*d^2 + a*e^2)^3*(d + e*x)*(a + c*x^2)^(3/2)) + (10*c*d*e^4*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)^(7/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(898\) vs. \(2(226)=452\).
time = 0.57, size = 899, normalized size = 3.68

method	result	size
default	\(\frac {-\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {5 c d e \left (\frac {e^{2}}{3 \left (e^{2} a +c \,d^{2}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {c d e \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}-\frac {4 c d}{3 e}}{\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{3 \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}+\frac {e^{2} \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}\right )}{e^{2} a +c \,d^{2}}-\frac {4 c \,e^{2} \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}-\frac {4 c d}{3 e}}{\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{3 \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}}{e^{2}}\)	\(899\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/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)^(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)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (224) = 448\).
time = 0.38, size = 551, normalized size = 2.26 \begin {gather*} \frac {5 \, c^{2} d^{2} x}{\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}} + \frac {5 \, c^{2} d^{2} x}{3 \, {\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 {10 \, c^{2} d^{2} x}{3 \, {\left (\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}\right )}} + \frac {5 \, c d}{\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}} + \frac {5 \, c d}{3 \, {\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 {4 \, c x}{3 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{2}\right )}} - \frac {8 \, c x}{3 \, {\left (\sqrt {c x^{2} + a} a^{2} c d^{2} + \sqrt {c x^{2} + a} a^{3} e^{2}\right )}} + \frac {5 \, c 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 (-3\right )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {7}{2}}} - \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{3} e^{\left (-1\right )} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{2} x + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a x e^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*c^2*d^2*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) + 5/3*c^2*d^2*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) + 10/3*c^2*d^2*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) + 5*c*d/(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*sqr
t(c*x^2 + a)*a^2*c*d^2*e + sqrt(c*x^2 + a)*a^3*e^3) + 5/3*c*d/((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) - 4/3*c*x/((c*x^2 + a)^(3/2)*a*c*d^2 + (c*x^2 + a)^(3/2)*a^2*e^
2) - 8/3*c*x/(sqrt(c*x^2 + a)*a^2*c*d^2 + sqrt(c*x^2 + a)*a^3*e^2) + 5*c*d*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/((c*x^2 + a)^(3/2)*c*d^3*e^(-1) + (c*x
^2 + a)^(3/2)*c*d^2*x + (c*x^2 + a)^(3/2)*a*x*e^2 + (c*x^2 + a)^(3/2)*a*d*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 815 vs. \(2 (224) = 448\).
time = 4.42, size = 1657, normalized size = 6.79 \begin {gather*} \left [\frac {15 \, \sqrt {c d^{2} + a e^{2}} {\left ({\left (a^{2} c^{3} d x^{5} + 2 \, a^{3} c^{2} d x^{3} + a^{4} c d x\right )} e^{5} + {\left (a^{2} c^{3} d^{2} x^{4} + 2 \, a^{3} c^{2} d^{2} x^{2} + a^{4} c d^{2}\right )} e^{4}\right )} \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 (2 \, c^{5} d^{7} x^{3} + 3 \, a c^{4} d^{7} x - {\left (8 \, a^{3} c^{2} x^{4} + 12 \, a^{4} c x^{2} + 3 \, a^{5}\right )} e^{7} + {\left (7 \, a^{3} c^{2} d x^{3} + 8 \, a^{4} c d x\right )} e^{6} + {\left (a^{2} c^{3} d^{2} x^{4} + 9 \, a^{3} c^{2} d^{2} x^{2} + 11 \, a^{4} c d^{2}\right )} e^{5} + {\left (16 \, a^{2} c^{3} d^{3} x^{3} + 19 \, a^{3} c^{2} d^{3} x\right )} e^{4} + {\left (11 \, a c^{4} d^{4} x^{4} + 24 \, a^{2} c^{3} d^{4} x^{2} + 16 \, a^{3} c^{2} d^{4}\right )} e^{3} + {\left (11 \, a c^{4} d^{5} x^{3} + 14 \, a^{2} c^{3} d^{5} x\right )} e^{2} + {\left (2 \, c^{5} d^{6} x^{4} + 3 \, a c^{4} d^{6} x^{2} + 2 \, a^{2} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{2} c^{6} d^{9} x^{4} + 2 \, a^{3} c^{5} d^{9} x^{2} + a^{4} c^{4} d^{9} + {\left (a^{6} c^{2} x^{5} + 2 \, a^{7} c x^{3} + a^{8} x\right )} e^{9} + {\left (a^{6} c^{2} d x^{4} + 2 \, a^{7} c d x^{2} + a^{8} d\right )} e^{8} + 4 \, {\left (a^{5} c^{3} d^{2} x^{5} + 2 \, a^{6} c^{2} d^{2} x^{3} + a^{7} c d^{2} x\right )} e^{7} + 4 \, {\left (a^{5} c^{3} d^{3} x^{4} + 2 \, a^{6} c^{2} d^{3} x^{2} + a^{7} c d^{3}\right )} e^{6} + 6 \, {\left (a^{4} c^{4} d^{4} x^{5} + 2 \, a^{5} c^{3} d^{4} x^{3} + a^{6} c^{2} d^{4} x\right )} e^{5} + 6 \, {\left (a^{4} c^{4} d^{5} x^{4} + 2 \, a^{5} c^{3} d^{5} x^{2} + a^{6} c^{2} d^{5}\right )} e^{4} + 4 \, {\left (a^{3} c^{5} d^{6} x^{5} + 2 \, a^{4} c^{4} d^{6} x^{3} + a^{5} c^{3} d^{6} x\right )} e^{3} + 4 \, {\left (a^{3} c^{5} d^{7} x^{4} + 2 \, a^{4} c^{4} d^{7} x^{2} + a^{5} c^{3} d^{7}\right )} e^{2} + {\left (a^{2} c^{6} d^{8} x^{5} + 2 \, a^{3} c^{5} d^{8} x^{3} + a^{4} c^{4} d^{8} x\right )} e\right )}}, \frac {15 \, \sqrt {-c d^{2} - a e^{2}} {\left ({\left (a^{2} c^{3} d x^{5} + 2 \, a^{3} c^{2} d x^{3} + a^{4} c d x\right )} e^{5} + {\left (a^{2} c^{3} d^{2} x^{4} + 2 \, a^{3} c^{2} d^{2} x^{2} + a^{4} c d^{2}\right )} e^{4}\right )} \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 (2 \, c^{5} d^{7} x^{3} + 3 \, a c^{4} d^{7} x - {\left (8 \, a^{3} c^{2} x^{4} + 12 \, a^{4} c x^{2} + 3 \, a^{5}\right )} e^{7} + {\left (7 \, a^{3} c^{2} d x^{3} + 8 \, a^{4} c d x\right )} e^{6} + {\left (a^{2} c^{3} d^{2} x^{4} + 9 \, a^{3} c^{2} d^{2} x^{2} + 11 \, a^{4} c d^{2}\right )} e^{5} + {\left (16 \, a^{2} c^{3} d^{3} x^{3} + 19 \, a^{3} c^{2} d^{3} x\right )} e^{4} + {\left (11 \, a c^{4} d^{4} x^{4} + 24 \, a^{2} c^{3} d^{4} x^{2} + 16 \, a^{3} c^{2} d^{4}\right )} e^{3} + {\left (11 \, a c^{4} d^{5} x^{3} + 14 \, a^{2} c^{3} d^{5} x\right )} e^{2} + {\left (2 \, c^{5} d^{6} x^{4} + 3 \, a c^{4} d^{6} x^{2} + 2 \, a^{2} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{6} d^{9} x^{4} + 2 \, a^{3} c^{5} d^{9} x^{2} + a^{4} c^{4} d^{9} + {\left (a^{6} c^{2} x^{5} + 2 \, a^{7} c x^{3} + a^{8} x\right )} e^{9} + {\left (a^{6} c^{2} d x^{4} + 2 \, a^{7} c d x^{2} + a^{8} d\right )} e^{8} + 4 \, {\left (a^{5} c^{3} d^{2} x^{5} + 2 \, a^{6} c^{2} d^{2} x^{3} + a^{7} c d^{2} x\right )} e^{7} + 4 \, {\left (a^{5} c^{3} d^{3} x^{4} + 2 \, a^{6} c^{2} d^{3} x^{2} + a^{7} c d^{3}\right )} e^{6} + 6 \, {\left (a^{4} c^{4} d^{4} x^{5} + 2 \, a^{5} c^{3} d^{4} x^{3} + a^{6} c^{2} d^{4} x\right )} e^{5} + 6 \, {\left (a^{4} c^{4} d^{5} x^{4} + 2 \, a^{5} c^{3} d^{5} x^{2} + a^{6} c^{2} d^{5}\right )} e^{4} + 4 \, {\left (a^{3} c^{5} d^{6} x^{5} + 2 \, a^{4} c^{4} d^{6} x^{3} + a^{5} c^{3} d^{6} x\right )} e^{3} + 4 \, {\left (a^{3} c^{5} d^{7} x^{4} + 2 \, a^{4} c^{4} d^{7} x^{2} + a^{5} c^{3} d^{7}\right )} e^{2} + {\left (a^{2} c^{6} d^{8} x^{5} + 2 \, a^{3} c^{5} d^{8} x^{3} + a^{4} c^{4} d^{8} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(15*sqrt(c*d^2 + a*e^2)*((a^2*c^3*d*x^5 + 2*a^3*c^2*d*x^3 + a^4*c*d*x)*e^5 + (a^2*c^3*d^2*x^4 + 2*a^3*c^2
*d^2*x^2 + a^4*c*d^2)*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)*s
qrt(c*x^2 + a) + (a*c*x^2 + 2*a^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(2*c^5*d^7*x^3 + 3*a*c^4*d^7*x - (8*a^3
*c^2*x^4 + 12*a^4*c*x^2 + 3*a^5)*e^7 + (7*a^3*c^2*d*x^3 + 8*a^4*c*d*x)*e^6 + (a^2*c^3*d^2*x^4 + 9*a^3*c^2*d^2*
x^2 + 11*a^4*c*d^2)*e^5 + (16*a^2*c^3*d^3*x^3 + 19*a^3*c^2*d^3*x)*e^4 + (11*a*c^4*d^4*x^4 + 24*a^2*c^3*d^4*x^2
+ 16*a^3*c^2*d^4)*e^3 + (11*a*c^4*d^5*x^3 + 14*a^2*c^3*d^5*x)*e^2 + (2*c^5*d^6*x^4 + 3*a*c^4*d^6*x^2 + 2*a^2*
c^3*d^6)*e)*sqrt(c*x^2 + a))/(a^2*c^6*d^9*x^4 + 2*a^3*c^5*d^9*x^2 + a^4*c^4*d^9 + (a^6*c^2*x^5 + 2*a^7*c*x^3 +
a^8*x)*e^9 + (a^6*c^2*d*x^4 + 2*a^7*c*d*x^2 + a^8*d)*e^8 + 4*(a^5*c^3*d^2*x^5 + 2*a^6*c^2*d^2*x^3 + a^7*c*d^2
*x)*e^7 + 4*(a^5*c^3*d^3*x^4 + 2*a^6*c^2*d^3*x^2 + a^7*c*d^3)*e^6 + 6*(a^4*c^4*d^4*x^5 + 2*a^5*c^3*d^4*x^3 + a
^6*c^2*d^4*x)*e^5 + 6*(a^4*c^4*d^5*x^4 + 2*a^5*c^3*d^5*x^2 + a^6*c^2*d^5)*e^4 + 4*(a^3*c^5*d^6*x^5 + 2*a^4*c^4
*d^6*x^3 + a^5*c^3*d^6*x)*e^3 + 4*(a^3*c^5*d^7*x^4 + 2*a^4*c^4*d^7*x^2 + a^5*c^3*d^7)*e^2 + (a^2*c^6*d^8*x^5 +
2*a^3*c^5*d^8*x^3 + a^4*c^4*d^8*x)*e), 1/3*(15*sqrt(-c*d^2 - a*e^2)*((a^2*c^3*d*x^5 + 2*a^3*c^2*d*x^3 + a^4*c
*d*x)*e^5 + (a^2*c^3*d^2*x^4 + 2*a^3*c^2*d^2*x^2 + a^4*c*d^2)*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)) + (2*c^5*d^7*x^3 + 3*a*c^4*d^7*x - (8*a^3*c^2*x
^4 + 12*a^4*c*x^2 + 3*a^5)*e^7 + (7*a^3*c^2*d*x^3 + 8*a^4*c*d*x)*e^6 + (a^2*c^3*d^2*x^4 + 9*a^3*c^2*d^2*x^2 +
11*a^4*c*d^2)*e^5 + (16*a^2*c^3*d^3*x^3 + 19*a^3*c^2*d^3*x)*e^4 + (11*a*c^4*d^4*x^4 + 24*a^2*c^3*d^4*x^2 + 16*
a^3*c^2*d^4)*e^3 + (11*a*c^4*d^5*x^3 + 14*a^2*c^3*d^5*x)*e^2 + (2*c^5*d^6*x^4 + 3*a*c^4*d^6*x^2 + 2*a^2*c^3*d^
6)*e)*sqrt(c*x^2 + a))/(a^2*c^6*d^9*x^4 + 2*a^3*c^5*d^9*x^2 + a^4*c^4*d^9 + (a^6*c^2*x^5 + 2*a^7*c*x^3 + a^8*x
)*e^9 + (a^6*c^2*d*x^4 + 2*a^7*c*d*x^2 + a^8*d)*e^8 + 4*(a^5*c^3*d^2*x^5 + 2*a^6*c^2*d^2*x^3 + a^7*c*d^2*x)*e^
7 + 4*(a^5*c^3*d^3*x^4 + 2*a^6*c^2*d^3*x^2 + a^7*c*d^3)*e^6 + 6*(a^4*c^4*d^4*x^5 + 2*a^5*c^3*d^4*x^3 + a^6*c^2
*d^4*x)*e^5 + 6*(a^4*c^4*d^5*x^4 + 2*a^5*c^3*d^5*x^2 + a^6*c^2*d^5)*e^4 + 4*(a^3*c^5*d^6*x^5 + 2*a^4*c^4*d^6*x
^3 + a^5*c^3*d^6*x)*e^3 + 4*(a^3*c^5*d^7*x^4 + 2*a^4*c^4*d^7*x^2 + a^5*c^3*d^7)*e^2 + (a^2*c^6*d^8*x^5 + 2*a^3
*c^5*d^8*x^3 + a^4*c^4*d^8*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 )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1073 vs. \(2 (224) = 448\).
time = 2.53, size = 1073, normalized size = 4.40 \begin {gather*} -\frac {1}{3} \, {\left (\frac {15 \, c d e^{7} \log \left ({\left | -c d + \sqrt {c d^{2} + a e^{2}} {\left (\sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} + a e^{4}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{{\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 )} \sqrt {c d^{2} + a e^{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {{\left (2 \, \sqrt {c d^{2} + a e^{2}} c^{3} d^{4} e^{2} + 9 \, \sqrt {c d^{2} + a e^{2}} a c^{2} d^{2} e^{4} - 15 \, a^{2} c^{\frac {3}{2}} d e^{6} \log \left ({\left | -c d + \sqrt {c d^{2} + a e^{2}} \sqrt {c} \right |}\right ) - 8 \, \sqrt {c d^{2} + a e^{2}} a^{2} c e^{6}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c d^{2} + a e^{2}} a^{2} c^{\frac {7}{2}} d^{6} + 3 \, \sqrt {c d^{2} + a e^{2}} a^{3} c^{\frac {5}{2}} d^{4} e^{2} + 3 \, \sqrt {c d^{2} + a e^{2}} a^{4} c^{\frac {3}{2}} d^{2} e^{4} + \sqrt {c d^{2} + a e^{2}} a^{5} \sqrt {c} e^{6}} + \frac {\frac {{\left (\frac {{\left (\frac {{\left (\frac {2 \, {\left (c^{5} d^{7} e^{16} + 6 \, a c^{4} d^{5} e^{18} - 11 \, a^{2} c^{3} d^{3} e^{20} - 16 \, a^{3} c^{2} d e^{22}\right )}}{a^{2} c^{4} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{3} c^{3} d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{4} c^{2} d^{2} e^{15} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{5} c e^{17} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {3 \, {\left (a^{2} c^{3} d^{4} e^{21} + 2 \, a^{3} c^{2} d^{2} e^{23} + a^{4} c e^{25}\right )} e^{\left (-1\right )}}{{\left (a^{2} c^{4} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{3} c^{3} d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{4} c^{2} d^{2} e^{15} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{5} c e^{17} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}}\right )} e^{\left (-1\right )}}{x e + d} - \frac {6 \, {\left (c^{5} d^{6} e^{15} + 5 \, a c^{4} d^{4} e^{17} - 8 \, a^{2} c^{3} d^{2} e^{19} - 2 \, a^{3} c^{2} e^{21}\right )}}{a^{2} c^{4} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{3} c^{3} d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{4} c^{2} d^{2} e^{15} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{5} c e^{17} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )}}{x e + d} + \frac {3 \, {\left (2 \, c^{5} d^{5} e^{14} + 9 \, a c^{4} d^{3} e^{16} - 13 \, a^{2} c^{3} d e^{18}\right )}}{a^{2} c^{4} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{3} c^{3} d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{4} c^{2} d^{2} e^{15} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{5} c e^{17} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )}}{x e + d} - \frac {2 \, c^{5} d^{4} e^{13} + 9 \, a c^{4} d^{2} e^{15} - 8 \, a^{2} c^{3} e^{17}}{a^{2} c^{4} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{3} c^{3} d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, a^{4} c^{2} d^{2} e^{15} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{5} c e^{17} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}}{{\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}^{\frac {3}{2}}}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(15*c*d*e^7*log(abs(-c*d + sqrt(c*d^2 + a*e^2)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e
+ d)^2) + sqrt(c*d^2*e^2 + a*e^4)*e^(-1)/(x*e + 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)*sqrt(c*d^2 + a*e^2)*sgn(1/(x*e + d))) + (2*sqrt(c*d^2 + a*e^2)*c^3*d^4*e^2 + 9*sqrt(c*d^2 + a*e^2)*a*c^2*d
^2*e^4 - 15*a^2*c^(3/2)*d*e^6*log(abs(-c*d + sqrt(c*d^2 + a*e^2)*sqrt(c))) - 8*sqrt(c*d^2 + a*e^2)*a^2*c*e^6)*
sgn(1/(x*e + d))/(sqrt(c*d^2 + a*e^2)*a^2*c^(7/2)*d^6 + 3*sqrt(c*d^2 + a*e^2)*a^3*c^(5/2)*d^4*e^2 + 3*sqrt(c*d
^2 + a*e^2)*a^4*c^(3/2)*d^2*e^4 + sqrt(c*d^2 + a*e^2)*a^5*sqrt(c)*e^6) + ((((2*(c^5*d^7*e^16 + 6*a*c^4*d^5*e^1
8 - 11*a^2*c^3*d^3*e^20 - 16*a^3*c^2*d*e^22)/(a^2*c^4*d^6*e^11*sgn(1/(x*e + d)) + 3*a^3*c^3*d^4*e^13*sgn(1/(x*
e + d)) + 3*a^4*c^2*d^2*e^15*sgn(1/(x*e + d)) + a^5*c*e^17*sgn(1/(x*e + d))) + 3*(a^2*c^3*d^4*e^21 + 2*a^3*c^2
*d^2*e^23 + a^4*c*e^25)*e^(-1)/((a^2*c^4*d^6*e^11*sgn(1/(x*e + d)) + 3*a^3*c^3*d^4*e^13*sgn(1/(x*e + d)) + 3*a
^4*c^2*d^2*e^15*sgn(1/(x*e + d)) + a^5*c*e^17*sgn(1/(x*e + d)))*(x*e + d)))*e^(-1)/(x*e + d) - 6*(c^5*d^6*e^15
+ 5*a*c^4*d^4*e^17 - 8*a^2*c^3*d^2*e^19 - 2*a^3*c^2*e^21)/(a^2*c^4*d^6*e^11*sgn(1/(x*e + d)) + 3*a^3*c^3*d^4*
e^13*sgn(1/(x*e + d)) + 3*a^4*c^2*d^2*e^15*sgn(1/(x*e + d)) + a^5*c*e^17*sgn(1/(x*e + d))))*e^(-1)/(x*e + d) +
3*(2*c^5*d^5*e^14 + 9*a*c^4*d^3*e^16 - 13*a^2*c^3*d*e^18)/(a^2*c^4*d^6*e^11*sgn(1/(x*e + d)) + 3*a^3*c^3*d^4*
e^13*sgn(1/(x*e + d)) + 3*a^4*c^2*d^2*e^15*sgn(1/(x*e + d)) + a^5*c*e^17*sgn(1/(x*e + d))))*e^(-1)/(x*e + d) -
(2*c^5*d^4*e^13 + 9*a*c^4*d^2*e^15 - 8*a^2*c^3*e^17)/(a^2*c^4*d^6*e^11*sgn(1/(x*e + d)) + 3*a^3*c^3*d^4*e^13*
sgn(1/(x*e + d)) + 3*a^4*c^2*d^2*e^15*sgn(1/(x*e + d)) + a^5*c*e^17*sgn(1/(x*e + d))))/(c - 2*c*d/(x*e + d) +
c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)^(3/2))*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 )}^2} \,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)^2),x)

[Out]

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

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