Optimal. Leaf size=574 \[ -\frac{8 \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (4 a^{2/3} e+5 \left (1-\sqrt{3}\right ) b^{2/3} c\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{15 \sqrt [4]{3} b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt [3]{a} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt{a+b x^3}}+\frac{8 c \sqrt{a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{4 d \sqrt{a+b x^3}}{3 b^2}+\frac{2 e x \sqrt{a+b x^3}}{5 b^2} \]
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Rubi [A] time = 0.46911, antiderivative size = 574, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1828, 1888, 1886, 261, 1878, 218, 1877} \[ -\frac{8 \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (4 a^{2/3} e+5 \left (1-\sqrt{3}\right ) b^{2/3} c\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{15 \sqrt [4]{3} b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt [3]{a} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt{a+b x^3}}+\frac{8 c \sqrt{a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{4 d \sqrt{a+b x^3}}{3 b^2}+\frac{2 e x \sqrt{a+b x^3}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 1828
Rule 1888
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x^4 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt{a+b x^3}}-\frac{2 \int \frac{a^2 e-2 a b c x-3 a b d x^2-\frac{3}{2} a b e x^3}{\sqrt{a+b x^3}} \, dx}{3 a b^2}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt{a+b x^3}}+\frac{2 e x \sqrt{a+b x^3}}{5 b^2}-\frac{4 \int \frac{4 a^2 b e-5 a b^2 c x-\frac{15}{2} a b^2 d x^2}{\sqrt{a+b x^3}} \, dx}{15 a b^3}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt{a+b x^3}}+\frac{2 e x \sqrt{a+b x^3}}{5 b^2}-\frac{4 \int \frac{4 a^2 b e-5 a b^2 c x}{\sqrt{a+b x^3}} \, dx}{15 a b^3}+\frac{(2 d) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx}{b}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt{a+b x^3}}+\frac{4 d \sqrt{a+b x^3}}{3 b^2}+\frac{2 e x \sqrt{a+b x^3}}{5 b^2}+\frac{(4 c) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{3 b^{4/3}}-\frac{\left (4 \sqrt [3]{a} \left (5 \left (1-\sqrt{3}\right ) b^{2/3} c+4 a^{2/3} e\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{15 b^2}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt{a+b x^3}}+\frac{4 d \sqrt{a+b x^3}}{3 b^2}+\frac{2 e x \sqrt{a+b x^3}}{5 b^2}+\frac{8 c \sqrt{a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt [3]{a} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{8 \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (5 \left (1-\sqrt{3}\right ) b^{2/3} c+4 a^{2/3} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{15 \sqrt [4]{3} b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.102964, size = 127, normalized size = 0.22 \[ \frac{2 \left (-15 b c x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right )-8 a e x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right )+10 a d+8 a e x+15 b c x^2+5 b d x^3+3 b e x^4\right )}{15 b^2 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 817, normalized size = 1.4 \begin{align*} \text{result too large to display} \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 (e x^{2} + d x + c\right )} x^{4}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{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 (e x^{6} + d x^{5} + c x^{4}\right )} \sqrt{b x^{3} + a}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.1381, size = 129, normalized size = 0.22 \begin{align*} d \left (\begin{cases} \frac{4 a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 x^{3}}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{c x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{8}{3}\right )} + \frac{e x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{10}{3}\right )} \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 (e x^{2} + d x + c\right )} x^{4}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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