Optimal. Leaf size=1330 \[ \text{result too large to display} \]
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Rubi [A] time = 2.00991, antiderivative size = 1330, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {50, 62, 623, 303, 218, 1877} \[ -\frac{15 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{(a+b x) (c+d x)} \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} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+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}} E\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 ) (b c-a d)^{8/3}}{28 \sqrt [3]{2} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (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}}+\frac{5\ 3^{3/4} \sqrt [3]{(a+b x) (c+d x)} \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} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+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 ) (b c-a d)^{8/3}}{7\ 2^{5/6} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (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}}+\frac{15 \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2} (b c-a d)^2}{14 \sqrt [3]{2} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \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 )}-\frac{15 (a+b x)^{2/3} (c+d x)^{2/3} (b c-a d)}{28 d^2}+\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 62
Rule 623
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/3}}{\sqrt [3]{c+d x}} \, dx &=\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d}-\frac{(5 (b c-a d)) \int \frac{(a+b x)^{2/3}}{\sqrt [3]{c+d x}} \, dx}{7 d}\\ &=-\frac{15 (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{28 d^2}+\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d}+\frac{\left (5 (b c-a d)^2\right ) \int \frac{1}{\sqrt [3]{a+b x} \sqrt [3]{c+d x}} \, dx}{14 d^2}\\ &=-\frac{15 (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{28 d^2}+\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d}+\frac{\left (5 (b c-a d)^2 \sqrt [3]{(a+b x) (c+d x)}\right ) \int \frac{1}{\sqrt [3]{a c+(b c+a d) x+b d x^2}} \, dx}{14 d^2 \sqrt [3]{a+b x} \sqrt [3]{c+d x}}\\ &=-\frac{15 (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{28 d^2}+\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d}+\frac{\left (15 (b c-a d)^2 \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\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 )}{14 d^2 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac{15 (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{28 d^2}+\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d}+\frac{\left (15 (b c-a d)^2 \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} x}{\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 )}{14\ 2^{2/3} \sqrt [3]{b} d^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}+\frac{\left (15 (b c-a d)^{8/3} \sqrt [3]{(a+b x) (c+d x)} \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 )}{14 \sqrt [6]{2} \sqrt{2+\sqrt{3}} \sqrt [3]{b} d^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac{15 (b c-a d) (a+b x)^{2/3} (c+d x)^{2/3}}{28 d^2}+\frac{3 (a+b x)^{5/3} (c+d x)^{2/3}}{7 d}+\frac{15 (b c-a d)^2 \sqrt [3]{(a+b x) (c+d x)} \sqrt{(b c+a d+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{14 \sqrt [3]{2} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \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 )}-\frac{15 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (b c-a d)^{8/3} \sqrt [3]{(a+b x) (c+d x)} \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}} E\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 )}{28 \sqrt [3]{2} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (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}}+\frac{5\ 3^{3/4} (b c-a d)^{8/3} \sqrt [3]{(a+b x) (c+d x)} \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 )}{7\ 2^{5/6} b^{2/3} d^{8/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (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*}
Mathematica [C] time = 0.0291162, size = 73, normalized size = 0.05 \[ \frac{3 (a+b x)^{8/3} \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{3},\frac{8}{3};\frac{11}{3};\frac{d (a+b x)}{a d-b c}\right )}{8 b \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{5}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{5}{3}}}{\sqrt [3]{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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