Optimal. Leaf size=568 \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} d^{2/3} ((a+b x) (c+d x))^{2/3} \sqrt {(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt {\frac {2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}}{2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}}\right ),-7-4 \sqrt {3}\right )}{2^{2/3} b^{4/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}-\frac {3 \sqrt [3]{c+d x}}{2 b (a+b x)^{2/3}} \]
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Rubi [A] time = 0.59, antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {47, 62, 623, 218} \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} d^{2/3} ((a+b x) (c+d x))^{2/3} \sqrt {(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt {\frac {2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{2^{2/3} b^{4/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}-\frac {3 \sqrt [3]{c+d x}}{2 b (a+b x)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 62
Rule 218
Rule 623
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{5/3}} \, dx &=-\frac {3 \sqrt [3]{c+d x}}{2 b (a+b x)^{2/3}}+\frac {d \int \frac {1}{(a+b x)^{2/3} (c+d x)^{2/3}} \, dx}{2 b}\\ &=-\frac {3 \sqrt [3]{c+d x}}{2 b (a+b x)^{2/3}}+\frac {\left (d ((a+b x) (c+d x))^{2/3}\right ) \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^{2/3}} \, dx}{2 b (a+b x)^{2/3} (c+d x)^{2/3}}\\ &=-\frac {3 \sqrt [3]{c+d x}}{2 b (a+b x)^{2/3}}+\frac {\left (3 d ((a+b x) (c+d x))^{2/3} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{2 b (a+b x)^{2/3} (c+d x)^{2/3} (b c+a d+2 b d x)}\\ &=-\frac {3 \sqrt [3]{c+d x}}{2 b (a+b x)^{2/3}}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} d^{2/3} ((a+b x) (c+d x))^{2/3} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{2^{2/3} b^{4/3} (a+b x)^{2/3} (c+d x)^{2/3} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 73, normalized size = 0.13 \[ -\frac {3 \sqrt [3]{c+d x} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {1}{3};\frac {d (a+b x)}{a d-b c}\right )}{2 b (a+b x)^{2/3} \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^{1/3}}{{\left (a+b\,x\right )}^{5/3}} \,d x \]
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 \[ \int \frac {\sqrt [3]{c + d x}}{\left (a + b x\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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