3.6.75 \(\int \frac {1}{(d+e x)^3 (a+c x^2)^{3/2}} \, dx\) [575]

Optimal. Leaf size=223 \[ \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {3 c e^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}} \]

[Out]

-3/2*c*e^2*(-a*e^2+4*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)^(7/2)+(c*d
*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)^2/(c*x^2+a)^(1/2)+1/2*e*(-3*a*e^2+2*c*d^2)*(c*x^2+a)^(1/2)/a/(a*e^2+c*d^2)^2/(
e*x+d)^2+1/2*c*d*e*(-13*a*e^2+2*c*d^2)*(c*x^2+a)^(1/2)/a/(a*e^2+c*d^2)^3/(e*x+d)

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Rubi [A]
time = 0.13, antiderivative size = 223, 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, 849, 821, 739, 212} \begin {gather*} \frac {c d e \sqrt {a+c x^2} \left (2 c d^2-13 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^3}+\frac {e \sqrt {a+c x^2} \left (2 c d^2-3 a e^2\right )}{2 a (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {3 c e^2 \left (4 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 )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*(d + e*x)^2*Sqrt[a + c*x^2]) + (e*(2*c*d^2 - 3*a*e^2)*Sqrt[a + c*x^2])/(2*a*(
c*d^2 + a*e^2)^2*(d + e*x)^2) + (c*d*e*(2*c*d^2 - 13*a*e^2)*Sqrt[a + c*x^2])/(2*a*(c*d^2 + a*e^2)^3*(d + e*x))
- (3*c*e^2*(4*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)
^(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 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 )^{3/2}} \, dx &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}-\frac {\int \frac {-3 a e^2-2 c d e x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {\int \frac {10 a c d e^2+c e \left (2 c d^2-3 a e^2\right ) x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {\left (3 c e^2 \left (4 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 )^3}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {\left (3 c e^2 \left (4 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 )^3}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (2 c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c d e \left (2 c d^2-13 a e^2\right ) \sqrt {a+c x^2}}{2 a \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {3 c e^2 \left (4 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 )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 10.34, size = 240, normalized size = 1.08 \begin {gather*} \frac {1}{2} \left (\frac {-a^3 e^5+2 c^3 d^3 x (d+e x)^2-a^2 c e^3 \left (10 d^2+11 d e x+3 e^2 x^2\right )+a c^2 d e \left (6 d^3+6 d^2 e x-14 d e^2 x^2-13 e^3 x^3\right )}{a \left (c d^2+a e^2\right )^3 (d+e x)^2 \sqrt {a+c x^2}}+\frac {3 c e^2 \left (4 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{7/2}}+\frac {3 c e^2 \left (-4 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 )^{7/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(a^3*e^5) + 2*c^3*d^3*x*(d + e*x)^2 - a^2*c*e^3*(10*d^2 + 11*d*e*x + 3*e^2*x^2) + a*c^2*d*e*(6*d^3 + 6*d^2*
e*x - 14*d*e^2*x^2 - 13*e^3*x^3))/(a*(c*d^2 + a*e^2)^3*(d + e*x)^2*Sqrt[a + c*x^2]) + (3*c*e^2*(4*c*d^2 - a*e^
2)*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) + (3*c*e^2*(-4*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqr
t[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(205)=410\).
time = 0.44, size = 933, normalized size = 4.18

method	result	size
default	\(\frac {-\frac {e^{2}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\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}}}}+\frac {5 c d e \left (-\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\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 {3 c d e \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}}-\frac {4 c \,e^{2} \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}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}-\frac {3 c \,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 )}{2 \left (e^{2} a +c \,d^{2}\right )}}{e^{3}}\)	\(933\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+a)^(3/2),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)^(1/2)+5/2*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)^(1/2)+3*c*d*e/(a*e^2+c*d
^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*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))-3/2*c/(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. 620 vs. \(2 (204) = 408\).
time = 0.35, size = 620, normalized size = 2.78 \begin {gather*} \frac {15 \, c^{3} d^{3} x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{3} d^{6} + 3 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} e^{2} + 3 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{4} + \sqrt {c x^{2} + a} a^{4} e^{6}\right )}} + \frac {15 \, c^{2} d^{2}}{2 \, {\left (\sqrt {c x^{2} + a} c^{3} d^{6} e^{\left (-1\right )} + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{4} e + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{3} + \sqrt {c x^{2} + a} a^{3} e^{5}\right )}} - \frac {13 \, c^{2} d x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{3} e^{4}\right )}} + \frac {15 \, 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 {7}{2}}} - \frac {5 \, c d}{2 \, {\left (\sqrt {c x^{2} + a} c^{2} d^{5} e^{\left (-1\right )} + \sqrt {c x^{2} + a} c^{2} d^{4} x + 2 \, \sqrt {c x^{2} + a} a c d^{2} x e^{2} + 2 \, \sqrt {c x^{2} + a} a c d^{3} e + \sqrt {c x^{2} + a} a^{2} x e^{4} + \sqrt {c x^{2} + a} a^{2} d e^{3}\right )}} - \frac {3 \, 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 {5}{2}}} - \frac {3 \, c}{2 \, {\left (\sqrt {c x^{2} + a} c^{2} d^{4} e^{\left (-1\right )} + 2 \, \sqrt {c x^{2} + a} a c d^{2} e + \sqrt {c x^{2} + a} a^{2} e^{3}\right )}} - \frac {1}{2 \, {\left (\sqrt {c x^{2} + a} c d^{2} x^{2} e + \sqrt {c x^{2} + a} c d^{4} e^{\left (-1\right )} + 2 \, \sqrt {c x^{2} + a} c d^{3} x + \sqrt {c x^{2} + a} a x^{2} e^{3} + 2 \, \sqrt {c x^{2} + a} a d x e^{2} + \sqrt {c x^{2} + a} 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)^(3/2),x, algorithm="maxima")

[Out]

15/2*c^3*d^3*x/(sqrt(c*x^2 + a)*a*c^3*d^6 + 3*sqrt(c*x^2 + a)*a^2*c^2*d^4*e^2 + 3*sqrt(c*x^2 + a)*a^3*c*d^2*e^
4 + sqrt(c*x^2 + a)*a^4*e^6) + 15/2*c^2*d^2/(sqrt(c*x^2 + a)*c^3*d^6*e^(-1) + 3*sqrt(c*x^2 + a)*a*c^2*d^4*e +
3*sqrt(c*x^2 + a)*a^2*c*d^2*e^3 + sqrt(c*x^2 + a)*a^3*e^5) - 13/2*c^2*d*x/(sqrt(c*x^2 + a)*a*c^2*d^4 + 2*sqrt(
c*x^2 + a)*a^2*c*d^2*e^2 + sqrt(c*x^2 + a)*a^3*e^4) + 15/2*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)^(7/2) - 5/2*c*d/(sqrt(c*x^2 + a)*c^2*d^5*e^(-1) + sqrt(c
*x^2 + a)*c^2*d^4*x + 2*sqrt(c*x^2 + a)*a*c*d^2*x*e^2 + 2*sqrt(c*x^2 + a)*a*c*d^3*e + sqrt(c*x^2 + a)*a^2*x*e^
4 + sqrt(c*x^2 + a)*a^2*d*e^3) - 3/2*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)^(5/2) - 3/2*c/(sqrt(c*x^2 + a)*c^2*d^4*e^(-1) + 2*sqrt(c*x^2 + a)*a*c*d^2*e + sqrt(c
*x^2 + a)*a^2*e^3) - 1/2/(sqrt(c*x^2 + a)*c*d^2*x^2*e + sqrt(c*x^2 + a)*c*d^4*e^(-1) + 2*sqrt(c*x^2 + a)*c*d^3
*x + sqrt(c*x^2 + a)*a*x^2*e^3 + 2*sqrt(c*x^2 + a)*a*d*x*e^2 + sqrt(c*x^2 + a)*a*d^2*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (204) = 408\).
time = 4.57, size = 1550, normalized size = 6.95 \begin {gather*} \left [-\frac {3 \, \sqrt {c d^{2} + a e^{2}} {\left ({\left (a^{2} c^{2} x^{4} + a^{3} c x^{2}\right )} e^{6} + 2 \, {\left (a^{2} c^{2} d x^{3} + a^{3} c d x\right )} e^{5} - {\left (4 \, a c^{3} d^{2} x^{4} + 3 \, a^{2} c^{2} d^{2} x^{2} - a^{3} c d^{2}\right )} e^{4} - 8 \, {\left (a c^{3} d^{3} x^{3} + a^{2} c^{2} d^{3} x\right )} e^{3} - 4 \, {\left (a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4}\right )} e^{2}\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^{4} d^{7} x - {\left (3 \, a^{3} c x^{2} + a^{4}\right )} e^{7} - {\left (13 \, a^{2} c^{2} d x^{3} + 11 \, a^{3} c d x\right )} e^{6} - {\left (17 \, a^{2} c^{2} d^{2} x^{2} + 11 \, a^{3} c d^{2}\right )} e^{5} - {\left (11 \, a c^{3} d^{3} x^{3} + 5 \, a^{2} c^{2} d^{3} x\right )} e^{4} - 2 \, {\left (5 \, a c^{3} d^{4} x^{2} + 2 \, a^{2} c^{2} d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{3} + 4 \, a c^{3} d^{5} x\right )} e^{2} + 2 \, {\left (2 \, c^{4} d^{6} x^{2} + 3 \, a c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{4 \, {\left (a c^{5} d^{10} x^{2} + a^{2} c^{4} d^{10} + {\left (a^{5} c x^{4} + a^{6} x^{2}\right )} e^{10} + 2 \, {\left (a^{5} c d x^{3} + a^{6} d x\right )} e^{9} + {\left (4 \, a^{4} c^{2} d^{2} x^{4} + 5 \, a^{5} c d^{2} x^{2} + a^{6} d^{2}\right )} e^{8} + 8 \, {\left (a^{4} c^{2} d^{3} x^{3} + a^{5} c d^{3} x\right )} e^{7} + 2 \, {\left (3 \, a^{3} c^{3} d^{4} x^{4} + 5 \, a^{4} c^{2} d^{4} x^{2} + 2 \, a^{5} c d^{4}\right )} e^{6} + 12 \, {\left (a^{3} c^{3} d^{5} x^{3} + a^{4} c^{2} d^{5} x\right )} e^{5} + 2 \, {\left (2 \, a^{2} c^{4} d^{6} x^{4} + 5 \, a^{3} c^{3} d^{6} x^{2} + 3 \, a^{4} c^{2} d^{6}\right )} e^{4} + 8 \, {\left (a^{2} c^{4} d^{7} x^{3} + a^{3} c^{3} d^{7} x\right )} e^{3} + {\left (a c^{5} d^{8} x^{4} + 5 \, a^{2} c^{4} d^{8} x^{2} + 4 \, a^{3} c^{3} d^{8}\right )} e^{2} + 2 \, {\left (a c^{5} d^{9} x^{3} + a^{2} c^{4} d^{9} x\right )} e\right )}}, -\frac {3 \, \sqrt {-c d^{2} - a e^{2}} {\left ({\left (a^{2} c^{2} x^{4} + a^{3} c x^{2}\right )} e^{6} + 2 \, {\left (a^{2} c^{2} d x^{3} + a^{3} c d x\right )} e^{5} - {\left (4 \, a c^{3} d^{2} x^{4} + 3 \, a^{2} c^{2} d^{2} x^{2} - a^{3} c d^{2}\right )} e^{4} - 8 \, {\left (a c^{3} d^{3} x^{3} + a^{2} c^{2} d^{3} x\right )} e^{3} - 4 \, {\left (a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4}\right )} e^{2}\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^{4} d^{7} x - {\left (3 \, a^{3} c x^{2} + a^{4}\right )} e^{7} - {\left (13 \, a^{2} c^{2} d x^{3} + 11 \, a^{3} c d x\right )} e^{6} - {\left (17 \, a^{2} c^{2} d^{2} x^{2} + 11 \, a^{3} c d^{2}\right )} e^{5} - {\left (11 \, a c^{3} d^{3} x^{3} + 5 \, a^{2} c^{2} d^{3} x\right )} e^{4} - 2 \, {\left (5 \, a c^{3} d^{4} x^{2} + 2 \, a^{2} c^{2} d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{3} + 4 \, a c^{3} d^{5} x\right )} e^{2} + 2 \, {\left (2 \, c^{4} d^{6} x^{2} + 3 \, a c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{5} d^{10} x^{2} + a^{2} c^{4} d^{10} + {\left (a^{5} c x^{4} + a^{6} x^{2}\right )} e^{10} + 2 \, {\left (a^{5} c d x^{3} + a^{6} d x\right )} e^{9} + {\left (4 \, a^{4} c^{2} d^{2} x^{4} + 5 \, a^{5} c d^{2} x^{2} + a^{6} d^{2}\right )} e^{8} + 8 \, {\left (a^{4} c^{2} d^{3} x^{3} + a^{5} c d^{3} x\right )} e^{7} + 2 \, {\left (3 \, a^{3} c^{3} d^{4} x^{4} + 5 \, a^{4} c^{2} d^{4} x^{2} + 2 \, a^{5} c d^{4}\right )} e^{6} + 12 \, {\left (a^{3} c^{3} d^{5} x^{3} + a^{4} c^{2} d^{5} x\right )} e^{5} + 2 \, {\left (2 \, a^{2} c^{4} d^{6} x^{4} + 5 \, a^{3} c^{3} d^{6} x^{2} + 3 \, a^{4} c^{2} d^{6}\right )} e^{4} + 8 \, {\left (a^{2} c^{4} d^{7} x^{3} + a^{3} c^{3} d^{7} x\right )} e^{3} + {\left (a c^{5} d^{8} x^{4} + 5 \, a^{2} c^{4} d^{8} x^{2} + 4 \, a^{3} c^{3} d^{8}\right )} e^{2} + 2 \, {\left (a c^{5} d^{9} x^{3} + a^{2} c^{4} d^{9} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(3*sqrt(c*d^2 + a*e^2)*((a^2*c^2*x^4 + a^3*c*x^2)*e^6 + 2*(a^2*c^2*d*x^3 + a^3*c*d*x)*e^5 - (4*a*c^3*d^2
*x^4 + 3*a^2*c^2*d^2*x^2 - a^3*c*d^2)*e^4 - 8*(a*c^3*d^3*x^3 + a^2*c^2*d^3*x)*e^3 - 4*(a*c^3*d^4*x^2 + a^2*c^2
*d^4)*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*(2*c^4*d^7*x - (3*a^3*c*x^2 + a^4)*e^7 - (13*a^2*c^2*d*
x^3 + 11*a^3*c*d*x)*e^6 - (17*a^2*c^2*d^2*x^2 + 11*a^3*c*d^2)*e^5 - (11*a*c^3*d^3*x^3 + 5*a^2*c^2*d^3*x)*e^4 -
2*(5*a*c^3*d^4*x^2 + 2*a^2*c^2*d^4)*e^3 + 2*(c^4*d^5*x^3 + 4*a*c^3*d^5*x)*e^2 + 2*(2*c^4*d^6*x^2 + 3*a*c^3*d^
6)*e)*sqrt(c*x^2 + a))/(a*c^5*d^10*x^2 + a^2*c^4*d^10 + (a^5*c*x^4 + a^6*x^2)*e^10 + 2*(a^5*c*d*x^3 + a^6*d*x)
*e^9 + (4*a^4*c^2*d^2*x^4 + 5*a^5*c*d^2*x^2 + a^6*d^2)*e^8 + 8*(a^4*c^2*d^3*x^3 + a^5*c*d^3*x)*e^7 + 2*(3*a^3*
c^3*d^4*x^4 + 5*a^4*c^2*d^4*x^2 + 2*a^5*c*d^4)*e^6 + 12*(a^3*c^3*d^5*x^3 + a^4*c^2*d^5*x)*e^5 + 2*(2*a^2*c^4*d
^6*x^4 + 5*a^3*c^3*d^6*x^2 + 3*a^4*c^2*d^6)*e^4 + 8*(a^2*c^4*d^7*x^3 + a^3*c^3*d^7*x)*e^3 + (a*c^5*d^8*x^4 + 5
*a^2*c^4*d^8*x^2 + 4*a^3*c^3*d^8)*e^2 + 2*(a*c^5*d^9*x^3 + a^2*c^4*d^9*x)*e), -1/2*(3*sqrt(-c*d^2 - a*e^2)*((a
^2*c^2*x^4 + a^3*c*x^2)*e^6 + 2*(a^2*c^2*d*x^3 + a^3*c*d*x)*e^5 - (4*a*c^3*d^2*x^4 + 3*a^2*c^2*d^2*x^2 - a^3*c
*d^2)*e^4 - 8*(a*c^3*d^3*x^3 + a^2*c^2*d^3*x)*e^3 - 4*(a*c^3*d^4*x^2 + a^2*c^2*d^4)*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)) - (2*c^4*d^7*x - (3*a^3*c
*x^2 + a^4)*e^7 - (13*a^2*c^2*d*x^3 + 11*a^3*c*d*x)*e^6 - (17*a^2*c^2*d^2*x^2 + 11*a^3*c*d^2)*e^5 - (11*a*c^3*
d^3*x^3 + 5*a^2*c^2*d^3*x)*e^4 - 2*(5*a*c^3*d^4*x^2 + 2*a^2*c^2*d^4)*e^3 + 2*(c^4*d^5*x^3 + 4*a*c^3*d^5*x)*e^2
+ 2*(2*c^4*d^6*x^2 + 3*a*c^3*d^6)*e)*sqrt(c*x^2 + a))/(a*c^5*d^10*x^2 + a^2*c^4*d^10 + (a^5*c*x^4 + a^6*x^2)*
e^10 + 2*(a^5*c*d*x^3 + a^6*d*x)*e^9 + (4*a^4*c^2*d^2*x^4 + 5*a^5*c*d^2*x^2 + a^6*d^2)*e^8 + 8*(a^4*c^2*d^3*x^
3 + a^5*c*d^3*x)*e^7 + 2*(3*a^3*c^3*d^4*x^4 + 5*a^4*c^2*d^4*x^2 + 2*a^5*c*d^4)*e^6 + 12*(a^3*c^3*d^5*x^3 + a^4
*c^2*d^5*x)*e^5 + 2*(2*a^2*c^4*d^6*x^4 + 5*a^3*c^3*d^6*x^2 + 3*a^4*c^2*d^6)*e^4 + 8*(a^2*c^4*d^7*x^3 + a^3*c^3
*d^7*x)*e^3 + (a*c^5*d^8*x^4 + 5*a^2*c^4*d^8*x^2 + 4*a^3*c^3*d^8)*e^2 + 2*(a*c^5*d^9*x^3 + a^2*c^4*d^9*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 {3}{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)**(3/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (204) = 408\).
time = 1.03, size = 649, normalized size = 2.91 \begin {gather*} \frac {\frac {{\left (c^{6} d^{9} - 6 \, a^{2} c^{4} d^{5} e^{4} - 8 \, a^{3} c^{3} d^{3} e^{6} - 3 \, a^{4} c^{2} d e^{8}\right )} x}{a c^{6} d^{12} + 6 \, a^{2} c^{5} d^{10} e^{2} + 15 \, a^{3} c^{4} d^{8} e^{4} + 20 \, a^{4} c^{3} d^{6} e^{6} + 15 \, a^{5} c^{2} d^{4} e^{8} + 6 \, a^{6} c d^{2} e^{10} + a^{7} e^{12}} + \frac {3 \, a c^{5} d^{8} e + 8 \, a^{2} c^{4} d^{6} e^{3} + 6 \, a^{3} c^{3} d^{4} e^{5} - a^{5} c e^{9}}{a c^{6} d^{12} + 6 \, a^{2} c^{5} d^{10} e^{2} + 15 \, a^{3} c^{4} d^{8} e^{4} + 20 \, a^{4} c^{3} d^{6} e^{6} + 15 \, a^{5} c^{2} d^{4} e^{8} + 6 \, a^{6} c d^{2} e^{10} + a^{7} e^{12}}}{\sqrt {c x^{2} + a}} + \frac {3 \, {\left (4 \, c^{2} d^{2} e^{2} - a c e^{4}\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^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} d^{3} e^{2} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} d^{2} e^{3} - 22 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} d^{2} e^{3} - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} d e^{4} - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c e^{5} + 7 \, a^{2} c^{\frac {3}{2}} d e^{4} - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c e^{5}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\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 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c^6*d^9 - 6*a^2*c^4*d^5*e^4 - 8*a^3*c^3*d^3*e^6 - 3*a^4*c^2*d*e^8)*x/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a
^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12) + (3*a*c^5*d^8*e + 8*a
^2*c^4*d^6*e^3 + 6*a^3*c^3*d^4*e^5 - a^5*c*e^9)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4
*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12))/sqrt(c*x^2 + a) + 3*(4*c^2*d^2*e^2 - a*c*e^4
)*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)) - (14*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*d^3*e^2 + 6*(sqrt
(c)*x - sqrt(c*x^2 + a))^3*c^2*d^2*e^3 - 22*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*d^2*e^3 - 7*(sqrt(c)*x - sqrt(
c*x^2 + a))^2*a*c^(3/2)*d*e^4 - (sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*e^5 + 7*a^2*c^(3/2)*d*e^4 - (sqrt(c)*x - s
qrt(c*x^2 + a))*a^2*c*e^5)/((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)*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 )}^{3/2}\,{\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)^(3/2)*(d + e*x)^3),x)

[Out]

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

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