Optimal. Leaf size=189 \[ -\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac {a (4 b c-7 a d) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{9/2}}-\frac {a \sqrt {c+d x^3} (4 b c-7 a d)}{3 b^4}-\frac {a \left (c+d x^3\right )^{3/2} (4 b c-7 a d)}{9 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b^2 d} \]
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Rubi [A] time = 0.24, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 89, 80, 50, 63, 208} \[ -\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}-\frac {a \left (c+d x^3\right )^{3/2} (4 b c-7 a d)}{9 b^3 (b c-a d)}-\frac {a \sqrt {c+d x^3} (4 b c-7 a d)}{3 b^4}+\frac {a (4 b c-7 a d) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{9/2}}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b^2 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 89
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^2} \, dx,x,x^3\right )\\ &=-\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {(c+d x)^{3/2} \left (-\frac {1}{2} a (2 b c-5 a d)+b (b c-a d) x\right )}{a+b x} \, dx,x,x^3\right )}{3 b^2 (b c-a d)}\\ &=\frac {2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac {(a (4 b c-7 a d)) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )}{6 b^2 (b c-a d)}\\ &=-\frac {a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac {(a (4 b c-7 a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^3\right )}{6 b^3}\\ &=-\frac {a (4 b c-7 a d) \sqrt {c+d x^3}}{3 b^4}-\frac {a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac {(a (4 b c-7 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 b^4}\\ &=-\frac {a (4 b c-7 a d) \sqrt {c+d x^3}}{3 b^4}-\frac {a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac {(a (4 b c-7 a d) (b c-a d)) \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^4 d}\\ &=-\frac {a (4 b c-7 a d) \sqrt {c+d x^3}}{3 b^4}-\frac {a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}+\frac {a (4 b c-7 a d) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 162, normalized size = 0.86 \[ \frac {\sqrt {c+d x^3} \left (105 a^3 d^2+5 a^2 b d \left (14 d x^3-19 c\right )+2 a b^2 \left (3 c^2-34 c d x^3-7 d^2 x^6\right )+6 b^3 x^3 \left (c+d x^3\right )^2\right )}{45 b^4 d \left (a+b x^3\right )}+\frac {a (4 b c-7 a d) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 443, normalized size = 2.34 \[ \left [-\frac {15 \, {\left (4 \, a^{2} b c d - 7 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 7 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) - 2 \, {\left (6 \, b^{3} d^{2} x^{9} + 2 \, {\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{6} + 6 \, a b^{2} c^{2} - 95 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \, {\left (3 \, b^{3} c^{2} - 34 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{90 \, {\left (b^{5} d x^{3} + a b^{4} d\right )}}, \frac {15 \, {\left (4 \, a^{2} b c d - 7 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 7 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (6 \, b^{3} d^{2} x^{9} + 2 \, {\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{6} + 6 \, a b^{2} c^{2} - 95 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \, {\left (3 \, b^{3} c^{2} - 34 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{45 \, {\left (b^{5} d x^{3} + a b^{4} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 211, normalized size = 1.12 \[ -\frac {{\left (4 \, a b^{2} c^{2} - 11 \, a^{2} b c d + 7 \, a^{3} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} b^{4}} - \frac {\sqrt {d x^{3} + c} a^{2} b c d - \sqrt {d x^{3} + c} a^{3} d^{2}}{3 \, {\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{4}} + \frac {2 \, {\left (3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} b^{8} d^{4} - 10 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} a b^{7} d^{5} - 30 \, \sqrt {d x^{3} + c} a b^{7} c d^{5} + 45 \, \sqrt {d x^{3} + c} a^{2} b^{6} d^{6}\right )}}{45 \, b^{10} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.37, size = 1003, normalized size = 5.31 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.75, size = 331, normalized size = 1.75 \[ \frac {\sqrt {d\,x^3+c}\,\left (\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4}+\frac {2\,c\,\left (\frac {2\,d\,\left (a\,d-2\,b\,c\right )}{b^3}+\frac {2\,a\,d^2}{b^3}+\frac {8\,c\,d}{5\,b^2}\right )}{3\,d}+\frac {2\,a\,\left (\frac {d\,\left (a\,d-2\,b\,c\right )}{b^3}+\frac {a\,d^2}{b^3}\right )}{b}\right )}{3\,d}+\frac {2\,d\,x^6\,\sqrt {d\,x^3+c}}{15\,b^2}-\frac {x^3\,\sqrt {d\,x^3+c}\,\left (\frac {2\,d\,\left (a\,d-2\,b\,c\right )}{b^3}+\frac {2\,a\,d^2}{b^3}+\frac {8\,c\,d}{5\,b^2}\right )}{9\,d}-\frac {a^2\,\left (\frac {2\,b\,c^2}{3\,\left (2\,b^2\,c-2\,a\,b\,d\right )}+\frac {a\,\left (\frac {2\,a\,d^2}{3\,\left (2\,b^2\,c-2\,a\,b\,d\right )}-\frac {4\,b\,c\,d}{3\,\left (2\,b^2\,c-2\,a\,b\,d\right )}\right )}{b}\right )\,\sqrt {d\,x^3+c}}{b^2\,\left (b\,x^3+a\right )}+\frac {a\,\ln \left (\frac {a\,d-2\,b\,c-b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\sqrt {a\,d-b\,c}\,\left (7\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,b^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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