Optimal. Leaf size=120 \[ \frac {2 a (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2}}-\frac {2 a \sqrt {c+d x^3} (b c-a d)}{3 b^3}-\frac {2 a \left (c+d x^3\right )^{3/2}}{9 b^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d} \]
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Rubi [A] time = 0.10, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 80, 50, 63, 208} \begin {gather*} -\frac {2 a \left (c+d x^3\right )^{3/2}}{9 b^2}-\frac {2 a \sqrt {c+d x^3} (b c-a d)}{3 b^3}+\frac {2 a (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2}}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x (c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )\\ &=\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d}-\frac {a \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )}{3 b}\\ &=-\frac {2 a \left (c+d x^3\right )^{3/2}}{9 b^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d}-\frac {(a (b c-a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac {2 a (b c-a d) \sqrt {c+d x^3}}{3 b^3}-\frac {2 a \left (c+d x^3\right )^{3/2}}{9 b^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d}-\frac {\left (a (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 b^3}\\ &=-\frac {2 a (b c-a d) \sqrt {c+d x^3}}{3 b^3}-\frac {2 a \left (c+d x^3\right )^{3/2}}{9 b^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d}-\frac {\left (2 a (b c-a d)^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 b^3 d}\\ &=-\frac {2 a (b c-a d) \sqrt {c+d x^3}}{3 b^3}-\frac {2 a \left (c+d x^3\right )^{3/2}}{9 b^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d}+\frac {2 a (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 111, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {c+d x^3} \left (15 a^2 d^2-5 a b d \left (4 c+d x^3\right )+3 b^2 \left (c+d x^3\right )^2\right )}{45 b^3 d}+\frac {2 a (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 138, normalized size = 1.15 \begin {gather*} \frac {2 \sqrt {c+d x^3} \left (15 a^2 d^2-20 a b c d-5 a b d^2 x^3+3 b^2 c^2+6 b^2 c d x^3+3 b^2 d^2 x^6\right )}{45 b^3 d}+\frac {2 a (a d-b c)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3} \sqrt {a d-b c}}{b c-a d}\right )}{3 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 297, normalized size = 2.48 \begin {gather*} \left [-\frac {15 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{6} + 3 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} + {\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{45 \, b^{3} d}, \frac {2 \, {\left (15 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (3 \, b^{2} d^{2} x^{6} + 3 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} + {\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, b^{3} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 151, normalized size = 1.26 \begin {gather*} -\frac {2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} b^{3}} + \frac {2 \, {\left (3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} b^{4} d^{4} - 5 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} a b^{3} d^{5} - 15 \, \sqrt {d x^{3} + c} a b^{3} c d^{5} + 15 \, \sqrt {d x^{3} + c} a^{2} b^{2} d^{6}\right )}}{45 \, b^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 531, normalized size = 4.42 \begin {gather*} -\frac {\left (\frac {2 \sqrt {d \,x^{3}+c}\, d \,x^{3}}{9 b}+\frac {2 \left (-\frac {2 c d}{3 b}-\frac {\left (a d -2 b c \right ) d}{b^{2}}\right ) \sqrt {d \,x^{3}+c}}{3 d}+\frac {i \left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \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 b^{2} d^{2} \left (a d -b c \right ) \sqrt {d \,x^{3}+c}}\right ) a}{b}+\frac {2 \left (d \,x^{3}+c \right )^{\frac {5}{2}}}{15 b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.13, size = 215, normalized size = 1.79 \begin {gather*} \frac {\sqrt {d\,x^3+c}\,\left (\frac {2\,c^2}{b}+\frac {2\,a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}+\frac {2\,c\,\left (\frac {2\,a\,d^2}{b^2}-\frac {12\,c\,d}{5\,b}\right )}{3\,d}\right )}{3\,d}+\frac {2\,d\,x^6\,\sqrt {d\,x^3+c}}{15\,b}-\frac {x^3\,\sqrt {d\,x^3+c}\,\left (\frac {2\,a\,d^2}{b^2}-\frac {12\,c\,d}{5\,b}\right )}{9\,d}+\frac {a\,\ln \left (\frac {a^2\,d^2+2\,b^2\,c^2-a\,b\,d^2\,x^3+b^2\,c\,d\,x^3-3\,a\,b\,c\,d+\sqrt {b}\,\sqrt {d\,x^3+c}\,{\left (a\,d-b\,c\right )}^{3/2}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,1{}\mathrm {i}}{3\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 70.60, size = 116, normalized size = 0.97 \begin {gather*} - \frac {2 a \left (c + d x^{3}\right )^{\frac {3}{2}}}{9 b^{2}} - \frac {2 a \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{3 b^{4} \sqrt {\frac {a d - b c}{b}}} + \frac {2 \left (c + d x^{3}\right )^{\frac {5}{2}}}{15 b d} + \frac {\sqrt {c + d x^{3}} \left (2 a^{2} d - 2 a b c\right )}{3 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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