Optimal. Leaf size=583 \[ -\frac {4 a e \sqrt {a+b x^3}}{9 b^2}+\frac {2 c x \sqrt {a+b x^3}}{5 b}+\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}-\frac {8 a d \sqrt {a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{7 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {4 \sqrt {2+\sqrt {3}} a \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{35 \sqrt [4]{3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.49, antiderivative size = 583, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1902, 1608,
1900, 267, 1892, 224, 1891} \begin {gather*} -\frac {4 \sqrt {2+\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 (7 \sqrt [3]{b} c-10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\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 )}{35 \sqrt [4]{3} 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 {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} d \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 )}{7 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 a d \sqrt {a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {4 a e \sqrt {a+b x^3}}{9 b^2}+\frac {2 c x \sqrt {a+b x^3}}{5 b}+\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 267
Rule 1608
Rule 1891
Rule 1892
Rule 1900
Rule 1902
Rubi steps
\begin {align*} \int \frac {x^3 \left (c+d x+e x^2\right )}{\sqrt {a+b x^3}} \, dx &=\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}+\frac {2 \int \frac {-3 a e x^2+\frac {9}{2} b c x^3+\frac {9}{2} b d x^4}{\sqrt {a+b x^3}} \, dx}{9 b}\\ &=\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}+\frac {2 \int \frac {x^2 \left (-3 a e+\frac {9 b c x}{2}+\frac {9}{2} b d x^2\right )}{\sqrt {a+b x^3}} \, dx}{9 b}\\ &=\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}+\frac {4 \int \frac {-9 a b d x-\frac {21}{2} a b e x^2+\frac {63}{4} b^2 c x^3}{\sqrt {a+b x^3}} \, dx}{63 b^2}\\ &=\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}+\frac {4 \int \frac {x \left (-9 a b d-\frac {21}{2} a b e x+\frac {63}{4} b^2 c x^2\right )}{\sqrt {a+b x^3}} \, dx}{63 b^2}\\ &=\frac {2 c x \sqrt {a+b x^3}}{5 b}+\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}+\frac {8 \int \frac {-\frac {63}{4} a b^2 c-\frac {45}{2} a b^2 d x-\frac {105}{4} a b^2 e x^2}{\sqrt {a+b x^3}} \, dx}{315 b^3}\\ &=\frac {2 c x \sqrt {a+b x^3}}{5 b}+\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}+\frac {8 \int \frac {-\frac {63}{4} a b^2 c-\frac {45}{2} a b^2 d x}{\sqrt {a+b x^3}} \, dx}{315 b^3}-\frac {(2 a e) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{3 b}\\ &=-\frac {4 a e \sqrt {a+b x^3}}{9 b^2}+\frac {2 c x \sqrt {a+b x^3}}{5 b}+\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}-\frac {(4 a d) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{7 b^{4/3}}-\frac {\left (2 a \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{35 b^{4/3}}\\ &=-\frac {4 a e \sqrt {a+b x^3}}{9 b^2}+\frac {2 c x \sqrt {a+b x^3}}{5 b}+\frac {2 d x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 e x^3 \sqrt {a+b x^3}}{9 b}-\frac {8 a d \sqrt {a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} d \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 )}{7 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 {4 \sqrt {2+\sqrt {3}} a \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\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 )}{35 \sqrt [4]{3} 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}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.10, size = 132, normalized size = 0.23 \begin {gather*} \frac {-2 \left (a+b x^3\right ) (70 a e-b x (63 c+5 x (9 d+7 e x)))-126 a b c 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 )-90 a b d x^2 \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{315 b^2 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 793, normalized size = 1.36 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 = 0.08, size = 81, normalized size = 0.14 \begin {gather*} -\frac {2 \, {\left (126 \, a \sqrt {b} c {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 180 \, a \sqrt {b} d {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (35 \, b e x^{3} + 45 \, b d x^{2} + 63 \, b c x - 70 \, a e\right )} \sqrt {b x^{3} + a}\right )}}{315 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.91, size = 129, normalized size = 0.22 \begin {gather*} e \left (\begin {cases} - \frac {4 a \sqrt {a + b x^{3}}}{9 b^{2}} + \frac {2 x^{3} \sqrt {a + b x^{3}}}{9 b} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 \sqrt {a}} & \text {otherwise} \end {cases}\right ) + \frac {c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {7}{3}\right )} + \frac {d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {8}{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^3\,\left (e\,x^2+d\,x+c\right )}{\sqrt {b\,x^3+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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