Optimal. Leaf size=125 \[ -\frac {2 a^2 \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2}}+\frac {2 a^2 \sqrt {c+d x^3}}{3 b^3}-\frac {2 \left (c+d x^3\right )^{3/2} (a d+b c)}{9 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d^2} \]
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Rubi [A] time = 0.13, antiderivative size = 125, 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, 88, 50, 63, 208} \begin {gather*} \frac {2 a^2 \sqrt {c+d x^3}}{3 b^3}-\frac {2 a^2 \sqrt {b c-a d} \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 )^{3/2} (a d+b c)}{9 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8 \sqrt {c+d x^3}}{a+b x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+d x}}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {(-b c-a d) \sqrt {c+d x}}{b^2 d}+\frac {a^2 \sqrt {c+d x}}{b^2 (a+b x)}+\frac {(c+d x)^{3/2}}{b d}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 (b c+a d) \left (c+d x^3\right )^{3/2}}{9 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^3\right )}{3 b^2}\\ &=\frac {2 a^2 \sqrt {c+d x^3}}{3 b^3}-\frac {2 (b c+a d) \left (c+d x^3\right )^{3/2}}{9 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d^2}+\frac {\left (a^2 (b c-a d)\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^2 \sqrt {c+d x^3}}{3 b^3}-\frac {2 (b c+a d) \left (c+d x^3\right )^{3/2}}{9 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d^2}+\frac {\left (2 a^2 (b c-a d)\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^2 \sqrt {c+d x^3}}{3 b^3}-\frac {2 (b c+a d) \left (c+d x^3\right )^{3/2}}{9 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d^2}-\frac {2 a^2 \sqrt {b c-a d} \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 = 0.31, size = 121, normalized size = 0.97 \begin {gather*} \frac {2 \sqrt {c+d x^3} \left (15 a^2 d^2-5 a b d \left (c+d x^3\right )+b^2 \left (-2 c^2+c d x^3+3 d^2 x^6\right )\right )}{45 b^3 d^2}-\frac {2 a^2 \sqrt {b c-a d} \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 = 0.16, size = 139, normalized size = 1.11 \begin {gather*} \frac {2 a^2 \sqrt {a d-b c} \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}}+\frac {2 \sqrt {c+d x^3} \left (15 a^2 d^2-5 a b c d-5 a b d^2 x^3-2 b^2 c^2+b^2 c d x^3+3 b^2 d^2 x^6\right )}{45 b^3 d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 280, normalized size = 2.24 \begin {gather*} \left [\frac {15 \, a^{2} d^{2} \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} - 2 \, b^{2} c^{2} - 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (b^{2} c d - 5 \, a b d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{45 \, b^{3} d^{2}}, -\frac {2 \, {\left (15 \, a^{2} d^{2} \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} - 2 \, b^{2} c^{2} - 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (b^{2} c d - 5 \, a b d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, b^{3} d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 139, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (a^{2} b c - a^{3} d\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^{8} - 5 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} b^{4} c d^{8} - 5 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} a b^{3} d^{9} + 15 \, \sqrt {d x^{3} + c} a^{2} b^{2} d^{10}\right )}}{45 \, b^{5} d^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.42, size = 514, normalized size = 4.11 \begin {gather*} \frac {\left (\frac {2 \sqrt {d \,x^{3}+c}}{3 b}+\frac {i \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 \,d^{2} \sqrt {d \,x^{3}+c}}\right ) a^{2}}{b^{2}}+\frac {\left (\frac {2 \sqrt {d \,x^{3}+c}\, x^{6}}{15}+\frac {2 \sqrt {d \,x^{3}+c}\, c \,x^{3}}{45 d}-\frac {4 \sqrt {d \,x^{3}+c}\, c^{2}}{45 d^{2}}\right ) b -\frac {2 \left (d \,x^{3}+c \right )^{\frac {3}{2}} a}{9 d}}{b^{2}} \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 = 6.17, size = 176, normalized size = 1.41 \begin {gather*} \frac {2\,a^2\,\sqrt {d\,x^3+c}}{3\,b^3}+\frac {2\,{\left (d\,x^3+c\right )}^{5/2}}{15\,b\,d^2}-\frac {2\,a\,{\left (d\,x^3+c\right )}^{3/2}}{9\,b^2\,d}-\frac {2\,c\,{\left (d\,x^3+c\right )}^{3/2}}{9\,b\,d^2}+\frac {a^2\,\ln \left (\frac {a^2\,d^2\,1{}\mathrm {i}+b^2\,c^2\,2{}\mathrm {i}-2\,\sqrt {b}\,\sqrt {d\,x^3+c}\,{\left (a\,d-b\,c\right )}^{3/2}-a\,b\,d^2\,x^3\,1{}\mathrm {i}+b^2\,c\,d\,x^3\,1{}\mathrm {i}-a\,b\,c\,d\,3{}\mathrm {i}}{2\,b\,x^3+2\,a}\right )\,\sqrt {a\,d-b\,c}\,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 = 30.27, size = 128, normalized size = 1.02 \begin {gather*} \frac {2 \left (\frac {a^{2} d^{3} \sqrt {c + d x^{3}}}{3 b^{3}} - \frac {a^{2} d^{3} \left (a d - b c\right ) \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 {d \left (c + d x^{3}\right )^{\frac {5}{2}}}{15 b} + \frac {\left (c + d x^{3}\right )^{\frac {3}{2}} \left (- a d^{2} - b c d\right )}{9 b^{2}}\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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