Optimal. Leaf size=574 \[ \frac {2 x \left (a e-b c x-b d x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {4 d \sqrt {a+b x^3}}{3 b^2}+\frac {2 e x \sqrt {a+b x^3}}{5 b^2}+\frac {8 c \sqrt {a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {4 \sqrt {2-\sqrt {3}} \sqrt [3]{a} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {8 \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (5 \left (1-\sqrt {3}\right ) b^{2/3} c+4 a^{2/3} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{15 \sqrt [4]{3} b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.32, 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 = {1842, 1902,
1900, 267, 1892, 224, 1891} \begin {gather*} -\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 (\text {ArcSin}\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 (\text {ArcSin}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 267
Rule 1842
Rule 1891
Rule 1892
Rule 1900
Rule 1902
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.07, size = 127, normalized size = 0.22 \begin {gather*} \frac {2 \left (10 a d+8 a e x+15 b c x^2+5 b d x^3+3 b e x^4-8 a e x \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a}\right )-15 b c x^2 \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{15 b^2 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 817, normalized size = 1.42 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 130, normalized size = 0.23 \begin {gather*} -\frac {2 \, {\left (16 \, {\left (a b e x^{3} + a^{2} e\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 20 \, {\left (b^{2} c x^{3} + a b c\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (3 \, b^{2} e x^{4} + 5 \, b^{2} d x^{3} - 5 \, b^{2} c x^{2} + 8 \, a b e x + 10 \, a b d\right )} \sqrt {b x^{3} + a}\right )}}{15 \, {\left (b^{4} x^{3} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.94, size = 129, normalized size = 0.22 \begin {gather*} 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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^4\,\left (e\,x^2+d\,x+c\right )}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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