Optimal. Leaf size=628 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right ) \left (\sqrt [3]{b} (2 a f+7 b c)+\left (1-\sqrt{3}\right ) \sqrt [3]{a} (4 a g+5 b d)\right )}{27 \sqrt [4]{3} a^2 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (4 a g+5 b d)}{27 a^2 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (4 a g+5 b d) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 (3 a e-x (x (4 a g+5 b d)+2 a f+7 b c))}{27 a^2 b \sqrt{a+b x^3}}+\frac{2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.500377, antiderivative size = 628, normalized size of antiderivative = 1., 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 1858
Rule 1854
Rule 1878
Rule 218
Rule 1877
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*}
Mathematica [C] time = 0.184539, 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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Maple [B] time = 0.006, size = 1673, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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