Optimal. Leaf size=399 \[ \frac {d \left (a+b x^3\right )^{4/3}}{4 c^2}+\frac {\left (a+b x^3\right )^{4/3} (4 b c-3 a d)}{12 a c^2}+\frac {\sqrt [3]{a+b x^3} (4 b c-3 a d)}{3 c^2}-\frac {\sqrt [3]{a+b x^3} (b c-a d)}{c^2}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^2 \sqrt [3]{d}}+\frac {\sqrt [3]{a} (4 b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 c^2}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 \sqrt [3]{d}}-\frac {\sqrt [3]{a} (4 b c-3 a d) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} c^2}-\frac {(b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^2 \sqrt [3]{d}}-\frac {\sqrt [3]{a} \log (x) (4 b c-3 a d)}{6 c^2}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3} \]
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Rubi [A] time = 0.48, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {446, 103, 156, 50, 57, 617, 204, 31, 58} \[ \frac {d \left (a+b x^3\right )^{4/3}}{4 c^2}+\frac {\left (a+b x^3\right )^{4/3} (4 b c-3 a d)}{12 a c^2}+\frac {\sqrt [3]{a+b x^3} (4 b c-3 a d)}{3 c^2}-\frac {\sqrt [3]{a+b x^3} (b c-a d)}{c^2}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^2 \sqrt [3]{d}}+\frac {\sqrt [3]{a} (4 b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 c^2}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 \sqrt [3]{d}}-\frac {\sqrt [3]{a} (4 b c-3 a d) \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} c^2}-\frac {(b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^2 \sqrt [3]{d}}-\frac {\sqrt [3]{a} \log (x) (4 b c-3 a d)}{6 c^2}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 58
Rule 103
Rule 156
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{4/3}}{x^4 \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{x^2 (c+d x)} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{4/3} \left (\frac {1}{3} (-4 b c+3 a d)-\frac {4 b d x}{3}\right )}{x (c+d x)} \, dx,x,x^3\right )}{3 a c}\\ &=-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}+\frac {d^2 \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )}{3 c^2}+\frac {(4 b c-3 a d) \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{x} \, dx,x,x^3\right )}{9 a c^2}\\ &=\frac {d \left (a+b x^3\right )^{4/3}}{4 c^2}+\frac {(4 b c-3 a d) \left (a+b x^3\right )^{4/3}}{12 a c^2}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}+\frac {(4 b c-3 a d) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x} \, dx,x,x^3\right )}{9 c^2}-\frac {(d (b c-a d)) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 c^2}\\ &=\frac {(4 b c-3 a d) \sqrt [3]{a+b x^3}}{3 c^2}-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{c^2}+\frac {d \left (a+b x^3\right )^{4/3}}{4 c^2}+\frac {(4 b c-3 a d) \left (a+b x^3\right )^{4/3}}{12 a c^2}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}+\frac {(a (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{9 c^2}+\frac {(b c-a d)^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 c^2}\\ &=\frac {(4 b c-3 a d) \sqrt [3]{a+b x^3}}{3 c^2}-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{c^2}+\frac {d \left (a+b x^3\right )^{4/3}}{4 c^2}+\frac {(4 b c-3 a d) \left (a+b x^3\right )^{4/3}}{12 a c^2}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}-\frac {\sqrt [3]{a} (4 b c-3 a d) \log (x)}{6 c^2}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^2 \sqrt [3]{d}}-\frac {\left (\sqrt [3]{a} (4 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 c^2}-\frac {\left (a^{2/3} (4 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 c^2}+\frac {(b c-a d)^{4/3} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c^2 \sqrt [3]{d}}+\frac {(b c-a d)^{5/3} \operatorname {Subst}\left (\int \frac {1}{\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c^2 d^{2/3}}\\ &=\frac {(4 b c-3 a d) \sqrt [3]{a+b x^3}}{3 c^2}-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{c^2}+\frac {d \left (a+b x^3\right )^{4/3}}{4 c^2}+\frac {(4 b c-3 a d) \left (a+b x^3\right )^{4/3}}{12 a c^2}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}-\frac {\sqrt [3]{a} (4 b c-3 a d) \log (x)}{6 c^2}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^2 \sqrt [3]{d}}+\frac {\sqrt [3]{a} (4 b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 c^2}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 \sqrt [3]{d}}+\frac {\left (\sqrt [3]{a} (4 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{3 c^2}+\frac {(b c-a d)^{4/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{c^2 \sqrt [3]{d}}\\ &=\frac {(4 b c-3 a d) \sqrt [3]{a+b x^3}}{3 c^2}-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{c^2}+\frac {d \left (a+b x^3\right )^{4/3}}{4 c^2}+\frac {(4 b c-3 a d) \left (a+b x^3\right )^{4/3}}{12 a c^2}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}-\frac {\sqrt [3]{a} (4 b c-3 a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^2}-\frac {(b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^2 \sqrt [3]{d}}-\frac {\sqrt [3]{a} (4 b c-3 a d) \log (x)}{6 c^2}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^2 \sqrt [3]{d}}+\frac {\sqrt [3]{a} (4 b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 c^2}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 \sqrt [3]{d}}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 389, normalized size = 0.97 \[ \frac {\frac {(4 b c-3 a d) \left (-\frac {1}{2} a^{4/3} \left (\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )\right )+3 a \sqrt [3]{a+b x^3}+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{3 c}+\frac {a \left (3 d^{4/3} \left (a+b x^3\right )^{4/3}-2 (b c-a d) \left (\sqrt [3]{b c-a d} \left (\log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt {3}}\right )\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )\right )}{4 c \sqrt [3]{d}}-\frac {\left (a+b x^3\right )^{7/3}}{x^3}}{3 a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.10, size = 383, normalized size = 0.96 \[ \frac {6 \, \sqrt {3} {\left (b c - a d\right )} x^{3} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 2 \, \sqrt {3} {\left (4 \, b c - 3 \, a d\right )} \left (-a\right )^{\frac {1}{3}} x^{3} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + {\left (4 \, b c - 3 \, a d\right )} \left (-a\right )^{\frac {1}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 3 \, {\left (b c - a d\right )} x^{3} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (4 \, b c - 3 \, a d\right )} \left (-a\right )^{\frac {1}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b c - a d\right )} x^{3} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a c}{18 \, c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 394, normalized size = 0.99 \[ -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {\sqrt {3} {\left (4 \, a^{\frac {1}{3}} b c - 3 \, a^{\frac {4}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, c^{2}} - \frac {{\left (4 \, a^{\frac {1}{3}} b c - 3 \, a^{\frac {4}{3}} d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, c^{2}} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, c^{2} d} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, c^{2} d} + \frac {{\left (4 \, a b c - 3 \, a^{2} d\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {2}{3}} c^{2}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} a}{3 \, c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{\left (d \,x^{3}+c \right ) x^{4}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.88, size = 2047, normalized size = 5.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{3}\right )^{\frac {4}{3}}}{x^{4} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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