Optimal. Leaf size=628 \[ \frac {\sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (4 a g+5 b d) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 \sqrt {a+b x^3} (4 a g+5 b d)}{27 a^2 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 (3 a e-x (x (4 a g+5 b d)+2 a f+7 b c))}{27 a^2 b \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (\sqrt [3]{b} (2 a f+7 b c)+\left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+5 b d)\right )}{27 \sqrt [4]{3} a^2 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.50, antiderivative size = 628, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1858, 1854, 1878, 218, 1877} \[ \frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (\sqrt [3]{b} (2 a f+7 b c)+\left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+5 b d)\right )}{27 \sqrt [4]{3} a^2 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 \sqrt {a+b x^3} (4 a g+5 b d)}{27 a^2 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (4 a g+5 b d) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 (3 a e-x (x (4 a g+5 b d)+2 a f+7 b c))}{27 a^2 b \sqrt {a+b x^3}}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 218
Rule 1854
Rule 1858
Rule 1877
Rule 1878
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {2 x \left (b c-a f+(b d-a g) x+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}-\frac {2 \int \frac {-\frac {1}{2} b (7 b c+2 a f)-\frac {1}{2} b (5 b d+4 a g) x-\frac {3}{2} b^2 e x^2}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a b^2}\\ &=\frac {2 x \left (b c-a f+(b d-a g) x+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}-\frac {2 (3 a e-x (7 b c+2 a f+(5 b d+4 a g) x))}{27 a^2 b \sqrt {a+b x^3}}+\frac {4 \int \frac {\frac {1}{4} b (7 b c+2 a f)-\frac {1}{4} b (5 b d+4 a g) x}{\sqrt {a+b x^3}} \, dx}{27 a^2 b^2}\\ &=\frac {2 x \left (b c-a f+(b d-a g) x+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}-\frac {2 (3 a e-x (7 b c+2 a f+(5 b d+4 a g) x))}{27 a^2 b \sqrt {a+b x^3}}-\frac {(5 b d+4 a g) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{27 a^2 b^{4/3}}+\frac {\left (\sqrt [3]{b} (7 b c+2 a f)+\left (1-\sqrt {3}\right ) \sqrt [3]{a} (5 b d+4 a g)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{27 a^2 b^{4/3}}\\ &=\frac {2 x \left (b c-a f+(b d-a g) x+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}-\frac {2 (5 b d+4 a g) \sqrt {a+b x^3}}{27 a^2 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 (3 a e-x (7 b c+2 a f+(5 b d+4 a g) x))}{27 a^2 b \sqrt {a+b x^3}}+\frac {\sqrt {2-\sqrt {3}} (5 b d+4 a g) \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 )}{9\ 3^{3/4} a^{5/3} 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 {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{b} (7 b c+2 a f)+\left (1-\sqrt {3}\right ) \sqrt [3]{a} (5 b d+4 a g)\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 )}{27 \sqrt [4]{3} a^2 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] time = 0.22, size = 170, normalized size = 0.27 \[ \frac {-4 a^2 (15 e+x (5 f+27 g x))+10 x \left (a+b x^3\right ) \sqrt {\frac {b x^3}{a}+1} (2 a f+7 b c) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a}\right )+40 a b x \left (5 c+f x^3\right )+27 x^2 \left (a+b x^3\right ) \sqrt {\frac {b x^3}{a}+1} (4 a g+5 b d) \, _2F_1\left (\frac {2}{3},\frac {5}{2};\frac {5}{3};-\frac {b x^3}{a}\right )+140 b^2 c x^4}{270 a^2 b \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]
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 \[ \int \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1673, normalized size = 2.66 \[ \text {result too large to display} \]
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 \[ \int \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {g\,x^4+f\,x^3+e\,x^2+d\,x+c}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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