Optimal. Leaf size=556 \[ -\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 d \sqrt {a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (91 \sqrt [3]{b} c-55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{5005 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {18 a \sqrt {a+b x^3} \left (91 c x+55 d x^2\right )}{5005}+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right ) \]
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Rubi [A] time = 0.33, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1852, 1853, 1878, 218, 1877} \[ \frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (91 \sqrt [3]{b} c-55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{5005 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 d \sqrt {a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {18 a \sqrt {a+b x^3} \left (91 c x+55 d x^2\right )}{5005}+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right ) \]
Antiderivative was successfully verified.
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Rule 218
Rule 1852
Rule 1853
Rule 1877
Rule 1878
Rubi steps
\begin {align*} \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx &=\int (c+d x) \left (a+b x^3\right )^{3/2} \, dx\\ &=\frac {2}{143} \left (13 c x+11 d x^2\right ) \left (a+b x^3\right )^{3/2}+\frac {1}{2} (9 a) \int \left (\frac {2 c}{11}+\frac {2 d x}{13}\right ) \sqrt {a+b x^3} \, dx\\ &=\frac {18 a \left (91 c x+55 d x^2\right ) \sqrt {a+b x^3}}{5005}+\frac {2}{143} \left (13 c x+11 d x^2\right ) \left (a+b x^3\right )^{3/2}+\frac {1}{4} \left (27 a^2\right ) \int \frac {\frac {4 c}{55}+\frac {4 d x}{91}}{\sqrt {a+b x^3}} \, dx\\ &=\frac {18 a \left (91 c x+55 d x^2\right ) \sqrt {a+b x^3}}{5005}+\frac {2}{143} \left (13 c x+11 d x^2\right ) \left (a+b x^3\right )^{3/2}+\frac {\left (27 a^2 d\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{91 \sqrt [3]{b}}+\frac {\left (27 a^2 \left (91 c-\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{5005}\\ &=\frac {54 a^2 d \sqrt {a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {18 a \left (91 c x+55 d x^2\right ) \sqrt {a+b x^3}}{5005}+\frac {2}{143} \left (13 c x+11 d x^2\right ) \left (a+b x^3\right )^{3/2}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} 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 )}{91 b^{2/3} \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 {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (91 \sqrt [3]{b} c-55 \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 )}{5005 b^{2/3} \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 = 0.03, size = 76, normalized size = 0.14 \[ \frac {a x \sqrt {a+b x^3} \left (2 c \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+d x \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{2 \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b d x^{4} + b c x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}, x\right ) \]
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 \[ \int {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1546, normalized size = 2.78 \[ \text {result too large to display} \]
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 \[ \int {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}\,{d x} \]
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 \[ \int \sqrt {b\,x^3+a}\,\left (b\,d\,x^4+b\,c\,x^3+a\,d\,x+a\,c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.36, size = 170, normalized size = 0.31 \[ \frac {a^{\frac {3}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {3}{2}} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\sqrt {a} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} b d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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