Optimal. Leaf size=574 \[ -\frac {8 \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (4 a^{2/3} e+5 \left (1-\sqrt {3}\right ) b^{2/3} c\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{15 \sqrt [4]{3} b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {4 \sqrt {2-\sqrt {3}} \sqrt [3]{a} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {8 c \sqrt {a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {4 d \sqrt {a+b x^3}}{3 b^2}+\frac {2 e x \sqrt {a+b x^3}}{5 b^2} \]
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Rubi [A] time = 0.47, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, 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 218
Rule 261
Rule 1828
Rule 1877
Rule 1878
Rule 1886
Rule 1888
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*}
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Mathematica [C] time = 0.13, 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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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x^{6} + d x^{5} + c x^{4}\right )} \sqrt {b x^{3} + a}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d x + c\right )} x^{4}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 817, normalized size = 1.42 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (e x^{2} + d x + c\right )} x^{4}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^4\,\left (e\,x^2+d\,x+c\right )}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.27, size = 129, normalized size = 0.22 \[ d \left (\begin {cases} \frac {4 a}{3 b^{2} \sqrt {a + b x^{3}}} + \frac {2 x^{3}}{3 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {c x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {8}{3}\right )} + \frac {e x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {10}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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