Optimal. Leaf size=114 \[ \frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a (b c-a d)^{3/2}}-\frac {2 d}{3 c \sqrt {c+d x^3} (b c-a d)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a c^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 85, 156, 63, 208} \[ \frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a (b c-a d)^{3/2}}-\frac {2 d}{3 c \sqrt {c+d x^3} (b c-a d)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 85
Rule 156
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {2 d}{3 c (b c-a d) \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {b c-a d-b d x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 c (b c-a d)}\\ &=-\frac {2 d}{3 c (b c-a d) \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a c}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a (b c-a d)}\\ &=-\frac {2 d}{3 c (b c-a d) \sqrt {c+d x^3}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a c d}-\frac {\left (2 b^2\right ) \operatorname {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 d (b c-a d)}\\ &=-\frac {2 d}{3 c (b c-a d) \sqrt {c+d x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a c^{3/2}}+\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a (b c-a d)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 89, normalized size = 0.78 \[ -\frac {2 \left (b c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \left (d x^3+c\right )}{b c-a d}\right )+(a d-b c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3}{c}+1\right )\right )}{3 a c \sqrt {c+d x^3} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.19, size = 790, normalized size = 6.93 \[ \left [-\frac {2 \, \sqrt {d x^{3} + c} a c d + {\left (b c^{2} d x^{3} + b c^{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 ) - {\left ({\left (b c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right )}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, \sqrt {d x^{3} + c} a c d - 2 \, {\left (b c^{2} d x^{3} + b c^{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 c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right )}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, \sqrt {d x^{3} + c} a c d - 2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (b c^{2} d x^{3} + b c^{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 )}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, {\left (\sqrt {d x^{3} + c} a c d - {\left (b c^{2} d x^{3} + b c^{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 c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right )\right )}}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 111, normalized size = 0.97 \[ -\frac {2 \, b^{2} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (a b c - a^{2} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, d}{3 \, \sqrt {d x^{3} + c} {\left (b c^{2} - a c d\right )}} + \frac {2 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a \sqrt {-c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.29, size = 512, normalized size = 4.49 \[ -\frac {\left (-\frac {i b \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (\textit {\_Z}^{3} b +a \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +a \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b +a \right ) d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}-\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {\left (2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +a \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +a \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b +a \right )\right ) b}{2 \left (a d -b c \right ) d}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) d}}\right )}{3 d^{2} \left (-a d +b c \right ) \left (a d -b c \right ) \sqrt {d \,x^{3}+c}}-\frac {2}{3 \left (a d -b c \right ) \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}\right ) b}{a}+\frac {-\frac {2 \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}+\frac {2}{3 \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}\, c}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )} {\left (d x^{3} + c\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.44, size = 139, normalized size = 1.22 \[ \frac {\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{3\,a\,c^{3/2}}+\frac {2\,d}{3\,c\,\sqrt {d\,x^3+c}\,\left (a\,d-b\,c\right )}+\frac {b^{3/2}\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,1{}\mathrm {i}}{3\,a\,{\left (a\,d-b\,c\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 24.63, size = 104, normalized size = 0.91 \[ \frac {2 d}{3 c \sqrt {c + d x^{3}} \left (a d - b c\right )} + \frac {2 b \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{3 a \sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {- c}} \right )}}{3 a c \sqrt {- c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________