Optimal. Leaf size=170 \[ \frac {\sqrt {c} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 \sqrt {b}}-\frac {\sqrt {c+d x^3} (2 b c-a d)}{3 a^2 \left (a+b x^3\right )}-\frac {c \sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.26, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 98, 151, 156, 63, 208} \begin {gather*} -\frac {\sqrt {c+d x^3} (2 b c-a d)}{3 a^2 \left (a+b x^3\right )}+\frac {\sqrt {c} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 \sqrt {b}}-\frac {c \sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 151
Rule 156
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx,x,x^3\right )\\ &=-\frac {c \sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} c (4 b c-3 a d)+\frac {1}{2} d (3 b c-2 a d) x}{x (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac {(2 b c-a d) \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {c \sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} c (4 b c-3 a d) (b c-a d)+\frac {1}{2} d (b c-a d) (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a^2 (b c-a d)}\\ &=-\frac {(2 b c-a d) \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {c \sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}-\frac {(c (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3}+\frac {((b c-a d) (4 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3}\\ &=-\frac {(2 b c-a d) \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {c \sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}-\frac {(c (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^3 d}+\frac {((b c-a d) (4 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 a^3 d}\\ &=-\frac {(2 b c-a d) \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {c \sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}+\frac {\sqrt {c} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 142, normalized size = 0.84 \begin {gather*} \frac {\frac {a \sqrt {c+d x^3} \left (-a c+a d x^3-2 b c x^3\right )}{x^3 \left (a+b x^3\right )}+\sqrt {c} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )-\frac {\sqrt {b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{\sqrt {b}}}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.68, size = 179, normalized size = 1.05 \begin {gather*} \frac {\left (4 b c^{3/2}-3 a \sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3}+\frac {\sqrt {c+d x^3} \left (-a c+a d x^3-2 b c x^3\right )}{3 a^2 x^3 \left (a+b x^3\right )}+\frac {\left (-a^2 d^2+5 a b c d-4 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3} \sqrt {a d-b c}}{b c-a d}\right )}{3 a^3 \sqrt {b} \sqrt {a d-b c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 838, normalized size = 4.93 \begin {gather*} \left [-\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{6} + {\left (4 \, a b c - a^{2} d\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 ) + {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{6} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, {\left ({\left (2 \, a b c - a^{2} d\right )} x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - a b d\right )} x^{6} + {\left (4 \, a b c - a^{2} d\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 ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{6} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, {\left ({\left (2 \, a b c - a^{2} d\right )} x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{6} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{6} + {\left (4 \, a b c - a^{2} d\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 ({\left (2 \, a b c - a^{2} d\right )} x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}, -\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{6} + {\left (4 \, a b c - a^{2} d\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 ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{6} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left ({\left (2 \, a b c - a^{2} d\right )} x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 216, normalized size = 1.27 \begin {gather*} \frac {{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} 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} a^{3}} - \frac {{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} b c d - 2 \, \sqrt {d x^{3} + c} b c^{2} d - {\left (d x^{3} + c\right )}^{\frac {3}{2}} a d^{2} + 2 \, \sqrt {d x^{3} + c} a c d^{2}}{3 \, {\left ({\left (d x^{3} + c\right )}^{2} b - 2 \, {\left (d x^{3} + c\right )} b c + b c^{2} + {\left (d x^{3} + c\right )} a d - a c d\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 1093, normalized size = 6.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{2} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.82, size = 531, normalized size = 3.12 \begin {gather*} \frac {\sqrt {c}\,\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 )\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^3}-\frac {c\,\sqrt {d\,x^3+c}}{3\,a^2\,x^3}-\frac {\sqrt {d\,x^3+c}\,\left (\frac {3\,a\,d-4\,b\,c}{2\,a^2}-\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {b\,d^2\,\left (a\,d+b\,c\right )}{a^3\,c^2}-\frac {a\,\left (\frac {b^2\,d^3}{2\,a^3\,c^2}-\frac {b^2\,d^3\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^2\,c^2\,\left (a^2\,d-a\,b\,c\right )}+\frac {b^2\,d^2\,\left (a\,d+b\,c\right )\,\left (3\,a\,d-4\,b\,c\right )}{3\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}+\frac {b\,\left (3\,a\,d-4\,b\,c\right )\,\left (a^2\,d^3+4\,a\,b\,c\,d^2-b^2\,c^2\,d\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {a^2\,d^3+4\,a\,b\,c\,d^2-b^2\,c^2\,d}{2\,a^3\,c^2}+\frac {b\,\left (3\,a\,d-4\,b\,c\right )\,\left (-4\,a^2\,c\,d^2+2\,a\,b\,c^2\,d+2\,b^2\,c^3\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {-4\,a^2\,c\,d^2+2\,a\,b\,c^2\,d+2\,b^2\,c^3}{2\,a^3\,c^2}+\frac {b\,{\left (3\,a\,d-4\,b\,c\right )}^2}{6\,a^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}\right )}{b\,x^3+a}+\frac {\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 )\,\sqrt {a\,d-b\,c}\,\left (a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,a^3\,\sqrt {b}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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