Optimal. Leaf size=161 \[ -\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 \sqrt {b c-a d}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 \sqrt {c}}-\frac {2 b \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 161, 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, 99, 151, 156, 63, 208} \begin {gather*} -\frac {2 b \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 \sqrt {b c-a d}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 \sqrt {c}}-\frac {\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 99
Rule 151
Rule 156
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (-4 b c+a d)-\frac {3 b d x}{2}}{x (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac {2 b \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} (b c-a d) (4 b c-a d)-b d (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 \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}+\frac {(b (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3}-\frac {(4 b c-a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3}\\ &=-\frac {2 b \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}+\frac {(b (4 b c-3 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 {(4 b c-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 {2 b \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 190, normalized size = 1.18 \begin {gather*} \frac {\sqrt {c} \left (a \left (a+2 b x^3\right ) \sqrt {c+d x^3} (b c-a d)+\sqrt {b} x^3 \left (a+b x^3\right ) (4 b c-3 a d) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )\right )-x^3 \left (a+b x^3\right ) \left (a^2 d^2-5 a b c d+4 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 \sqrt {c} x^3 \left (a+b x^3\right ) (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 157, normalized size = 0.98 \begin {gather*} \frac {\left (3 a \sqrt {b} d-4 b^{3/2} c\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 {a d-b c}}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 \sqrt {c}}+\frac {\left (-a-2 b x^3\right ) \sqrt {c+d x^3}}{3 a^2 x^3 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 870, normalized size = 5.40 \begin {gather*} \left [-\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{6} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\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 {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, {\left (2 \, a b c x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b c x^{6} + a^{4} c x^{3}\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{6} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b 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 {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, {\left (2 \, a b c x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b c x^{6} + a^{4} c 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 {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{6} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right ) + 2 \, {\left (2 \, a b c x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b c x^{6} + a^{4} c x^{3}\right )}}, -\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{6} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b 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 {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (2 \, a b c x^{3} + a^{2} c\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a^{3} b c x^{6} + a^{4} c 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 = 183, normalized size = 1.14 \begin {gather*} \frac {{\left (4 \, b^{2} c - 3 \, a b 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} a^{3}} - \frac {{\left (4 \, b c - a 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 d - 2 \, \sqrt {d x^{3} + c} b c d + \sqrt {d x^{3} + c} a 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 = 0.26, size = 978, normalized size = 6.07
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 {\sqrt {d x^{3} + c}}{{\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 = 9.69, size = 438, normalized size = 2.72 \begin {gather*} \frac {\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {b^2\,d^2}{2\,a^3\,c^2}-\frac {b^2\,d^2\,\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\,\left (2\,a\,d-b\,c\right )\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {b\,d\,\left (2\,a\,d-b\,c\right )}{2\,a^3\,c^2}+\frac {b\,\left (3\,a\,d-4\,b\,c\right )\,\left (-a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {-a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2}{2\,a^3\,c^2}+\frac {b\,\left (a\,d-4\,b\,c\right )\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^2\,c\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {a\,d-4\,b\,c}{2\,a^2\,c}\right )\,\sqrt {d\,x^3+c}}{b\,x^3+a}-\frac {\sqrt {d\,x^3+c}}{3\,a^2\,x^3}+\frac {\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 (a\,d-4\,b\,c\right )}{6\,a^3\,\sqrt {c}}+\frac {\sqrt {b}\,\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 )\,\left (3\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,a^3\,\sqrt {a\,d-b\,c}} \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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