3.5.93 \(\int \frac {1}{x (a+b x^3)^2 (c+d x^3)^{3/2}} \, dx\) [493]

Optimal. Leaf size=172 \[ \frac {d (b c+2 a d)}{3 a c (b c-a d)^2 \sqrt {c+d x^3}}+\frac {b}{3 a (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2 c^{3/2}}+\frac {b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 (b c-a d)^{5/2}} \]

[Out]

-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))/a^2/c^(3/2)+1/3*b^(3/2)*(-5*a*d+2*b*c)*arctanh(b^(1/2)*(d*x^3+c)^(1/2)/(
-a*d+b*c)^(1/2))/a^2/(-a*d+b*c)^(5/2)+1/3*d*(2*a*d+b*c)/a/c/(-a*d+b*c)^2/(d*x^3+c)^(1/2)+1/3*b/a/(-a*d+b*c)/(b
*x^3+a)/(d*x^3+c)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 105, 157, 162, 65, 214} \begin {gather*} \frac {b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 (b c-a d)^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2 c^{3/2}}+\frac {b}{3 a \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}+\frac {d (2 a d+b c)}{3 a c \sqrt {c+d x^3} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*c + 2*a*d))/(3*a*c*(b*c - a*d)^2*Sqrt[c + d*x^3]) + b/(3*a*(b*c - a*d)*(a + b*x^3)*Sqrt[c + d*x^3]) - (2
*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2*c^(3/2)) + (b^(3/2)*(2*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3]
)/Sqrt[b*c - a*d]])/(3*a^2*(b*c - a*d)^(5/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

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

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x (a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {b}{3 a (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {\text {Subst}\left (\int \frac {b c-a d+\frac {3 b d x}{2}}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{3 a (b c-a d)}\\ &=\frac {d (b c+2 a d)}{3 a c (b c-a d)^2 \sqrt {c+d x^3}}+\frac {b}{3 a (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{2} (b c-a d)^2-\frac {1}{4} b d (b c+2 a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a c (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{3 a c (b c-a d)^2 \sqrt {c+d x^3}}+\frac {b}{3 a (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a^2 c}-\frac {\left (b^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^2 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{3 a c (b c-a d)^2 \sqrt {c+d x^3}}+\frac {b}{3 a (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^2 c d}-\frac {\left (b^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^2 d (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{3 a c (b c-a d)^2 \sqrt {c+d x^3}}+\frac {b}{3 a (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2 c^{3/2}}+\frac {b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 (b c-a d)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.76, size = 157, normalized size = 0.91 \begin {gather*} \frac {\frac {a \left (2 a^2 d^2+2 a b d^2 x^3+b^2 c \left (c+d x^3\right )\right )}{c (b c-a d)^2 \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {b^{3/2} (2 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{c^{3/2}}}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(2*a^2*d^2 + 2*a*b*d^2*x^3 + b^2*c*(c + d*x^3)))/(c*(b*c - a*d)^2*(a + b*x^3)*Sqrt[c + d*x^3]) - (b^(3/2)*
(2*b*c - 5*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(5/2) - (2*ArcTanh[Sqrt[c
+ d*x^3]/Sqrt[c]])/c^(3/2))/(3*a^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.45, size = 1002, normalized size = 5.83

method	result	size
default	\(\text {Expression too large to display}\)	\(1002\)
elliptic	\(\text {Expression too large to display}\)	\(1720\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/a^2*(-2/3/(a*d-b*c)/((x^3+c/d)*d)^(1/2)-1/3*I*b/d^2*2^(1/2)*sum(1/(-a*d+b*c)/(a*d-b*c)*(-c*d^2)^(1/3)*(1/2*
I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(
-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(
-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d^
2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-
c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alpha^2*d-I*(-c*d^2)^(2/3)*
3^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*
d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b+a)))-b/a*(-1/3*b/(a*d-b*c)^2*(d*x^3+c)
^(1/2)/(b*x^3+a)-2/3*d/(a*d-b*c)^2/((x^3+c/d)*d)^(1/2)+1/2*I*b/d*2^(1/2)*sum(1/(a*d-b*c)^3*(-c*d^2)^(1/3)*(1/2
*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*
(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/
(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d
^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(
-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alpha^2*d-I*(-c*d^2)^(2/3)
*3^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c
*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b+a)))+1/a^2*(2/3/c/((x^3+c/d)*d)^(1/2)
-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (144) = 288\).
time = 3.70, size = 1819, normalized size = 10.58 \begin {gather*} \left [-\frac {{\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{6} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right ) - 2 \, {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 2 \, {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{3}\right )}}, \frac {{\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{6} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b c}\right ) + {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{3}\right )}}, \frac {4 \, {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - {\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{6} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right ) + 2 \, {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{3}\right )}}, \frac {{\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{6} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b c}\right ) + 2 \, {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*((2*a*b^2*c^4 - 5*a^2*b*c^3*d + (2*b^3*c^3*d - 5*a*b^2*c^2*d^2)*x^6 + (2*b^3*c^4 - 3*a*b^2*c^3*d - 5*a^2
*b*c^2*d^2)*x^3)*sqrt(b/(b*c - a*d))*log((b*d*x^3 + 2*b*c - a*d - 2*sqrt(d*x^3 + c)*(b*c - a*d)*sqrt(b/(b*c -
a*d)))/(b*x^3 + a)) - 2*((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^6 + a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 +
(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^3)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^
3) - 2*(a*b^2*c^3 + 2*a^3*c*d^2 + (a*b^2*c^2*d + 2*a^2*b*c*d^2)*x^3)*sqrt(d*x^3 + c))/(a^3*b^2*c^5 - 2*a^4*b*c
^4*d + a^5*c^3*d^2 + (a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^6 + (a^2*b^3*c^5 - a^3*b^2*c^4*d -
a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3), 1/3*((2*a*b^2*c^4 - 5*a^2*b*c^3*d + (2*b^3*c^3*d - 5*a*b^2*c^2*d^2)*x^6 + (
2*b^3*c^4 - 3*a*b^2*c^3*d - 5*a^2*b*c^2*d^2)*x^3)*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^3 + c)*(b*c - a*d)*sqr
t(-b/(b*c - a*d))/(b*d*x^3 + b*c)) + ((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^6 + a*b^2*c^3 - 2*a^2*b*c^2*d
+ a^3*c*d^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^3)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt
(c) + 2*c)/x^3) + (a*b^2*c^3 + 2*a^3*c*d^2 + (a*b^2*c^2*d + 2*a^2*b*c*d^2)*x^3)*sqrt(d*x^3 + c))/(a^3*b^2*c^5
- 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^6 + (a^2*b^3*c^5 - a^3*b
^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3), 1/6*(4*((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^6 + a*b^2*c^3
- 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^3)*sqrt(-c)*arctan(sqrt(d*x^3
+ c)*sqrt(-c)/c) - (2*a*b^2*c^4 - 5*a^2*b*c^3*d + (2*b^3*c^3*d - 5*a*b^2*c^2*d^2)*x^6 + (2*b^3*c^4 - 3*a*b^2*c
^3*d - 5*a^2*b*c^2*d^2)*x^3)*sqrt(b/(b*c - a*d))*log((b*d*x^3 + 2*b*c - a*d - 2*sqrt(d*x^3 + c)*(b*c - a*d)*sq
rt(b/(b*c - a*d)))/(b*x^3 + a)) + 2*(a*b^2*c^3 + 2*a^3*c*d^2 + (a*b^2*c^2*d + 2*a^2*b*c*d^2)*x^3)*sqrt(d*x^3 +
c))/(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^6 + (a
^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3), 1/3*((2*a*b^2*c^4 - 5*a^2*b*c^3*d + (2*b^3*c^3
*d - 5*a*b^2*c^2*d^2)*x^6 + (2*b^3*c^4 - 3*a*b^2*c^3*d - 5*a^2*b*c^2*d^2)*x^3)*sqrt(-b/(b*c - a*d))*arctan(-sq
rt(d*x^3 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^3 + b*c)) + 2*((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x
^6 + a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^3)*sqrt(-c)*arc
tan(sqrt(d*x^3 + c)*sqrt(-c)/c) + (a*b^2*c^3 + 2*a^3*c*d^2 + (a*b^2*c^2*d + 2*a^2*b*c*d^2)*x^3)*sqrt(d*x^3 + c
))/(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^6 + (a^2
*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.28, size = 226, normalized size = 1.31 \begin {gather*} -\frac {{\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {{\left (d x^{3} + c\right )} b^{2} c d + 2 \, {\left (d x^{3} + c\right )} a b d^{2} - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}}{3 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{3} + c} b c + \sqrt {d x^{3} + c} a d\right )}} + \frac {2 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{2} \sqrt {-c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/((a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^
2)*sqrt(-b^2*c + a*b*d)) + 1/3*((d*x^3 + c)*b^2*c*d + 2*(d*x^3 + c)*a*b*d^2 - 2*a*b*c*d^2 + 2*a^2*d^3)/((a*b^2
*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*((d*x^3 + c)^(3/2)*b - sqrt(d*x^3 + c)*b*c + sqrt(d*x^3 + c)*a*d)) + 2/3*arc
tan(sqrt(d*x^3 + c)/sqrt(-c))/(a^2*sqrt(-c)*c)

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Mupad [B]
time = 12.18, size = 288, normalized size = 1.67 \begin {gather*} \frac {\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{3\,a^2\,c^{3/2}}+\frac {\left (\frac {{\left (2\,a\,d+b\,c\right )}^4+{\left (2\,a\,d+b\,c\right )}^2\,\left (\left (a\,d+2\,b\,c\right )\,\left (2\,a\,d+b\,c\right )-9\,a\,b\,c\,d\right )}{9\,a\,c\,{\left (2\,a\,d+b\,c\right )}^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b\,d\,x^3\,\left (2\,a\,d+b\,c\right )}{3\,a\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,\sqrt {d\,x^3+c}}{b\,d\,x^6+\left (a\,d+b\,c\right )\,x^3+a\,c}+\frac {b^{3/2}\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (5\,a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{6\,a^2\,{\left (a\,d-b\,c\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((((c + d*x^3)^(1/2) - c^(1/2))^3*((c + d*x^3)^(1/2) + c^(1/2)))/x^6)/(3*a^2*c^(3/2)) + ((((2*a*d + b*c)^4
+ (2*a*d + b*c)^2*((a*d + 2*b*c)*(2*a*d + b*c) - 9*a*b*c*d))/(9*a*c*(2*a*d + b*c)^2*(a^2*d^2 + b^2*c^2 - 2*a*b
*c*d)) + (b*d*x^3*(2*a*d + b*c))/(3*a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))*(c + d*x^3)^(1/2))/(a*c + x^3*(a*d +
b*c) + b*d*x^6) + (b^(3/2)*log((2*b*c - a*d + b^(1/2)*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2)*2i + b*d*x^3)/(a +
b*x^3))*(5*a*d - 2*b*c)*1i)/(6*a^2*(a*d - b*c)^(5/2))

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