Optimal. Leaf size=125 \[ \frac {2 a^2 \sqrt {c+d x^3}}{3 b^3}-\frac {2 (b c+a d) \left (c+d x^3\right )^{3/2}}{9 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{5/2}}{15 b d^2}-\frac {2 a^2 \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2}} \]
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Rubi [A]
time = 0.09, 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 = {457, 90, 52, 65,
214} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {x^8 \sqrt {c+d x^3}}{a+b x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2 \sqrt {c+d x}}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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 \text {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 ) \text {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 ) \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 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 = 0.22, 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} \tan ^{-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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.38, size = 514, normalized size = 4.11 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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.23, 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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Sympy [A]
time = 12.26, 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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Giac [A]
time = 0.52, 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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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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