Optimal. Leaf size=299 \[ \frac {e^2 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{27 \sqrt [4]{3} a^{4/3} b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 e^2 \sqrt {e x} (7 a B+2 A b)}{27 a b^2 \sqrt {a+b x^3}}+\frac {2 (e x)^{7/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {457, 288, 329, 225} \[ \frac {e^2 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (7 a B+2 A b) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{27 \sqrt [4]{3} a^{4/3} b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 e^2 \sqrt {e x} (7 a B+2 A b)}{27 a b^2 \sqrt {a+b x^3}}+\frac {2 (e x)^{7/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 225
Rule 288
Rule 329
Rule 457
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {2 (A b-a B) (e x)^{7/2}}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac {\left (2 \left (A b+\frac {7 a B}{2}\right )\right ) \int \frac {(e x)^{5/2}}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a b}\\ &=\frac {2 (A b-a B) (e x)^{7/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 (2 A b+7 a B) e^2 \sqrt {e x}}{27 a b^2 \sqrt {a+b x^3}}+\frac {\left ((2 A b+7 a B) e^3\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^3}} \, dx}{27 a b^2}\\ &=\frac {2 (A b-a B) (e x)^{7/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 (2 A b+7 a B) e^2 \sqrt {e x}}{27 a b^2 \sqrt {a+b x^3}}+\frac {\left (2 (2 A b+7 a B) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{27 a b^2}\\ &=\frac {2 (A b-a B) (e x)^{7/2}}{9 a b e \left (a+b x^3\right )^{3/2}}-\frac {2 (2 A b+7 a B) e^2 \sqrt {e x}}{27 a b^2 \sqrt {a+b x^3}}+\frac {(2 A b+7 a B) e^2 \sqrt {e x} \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 (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{27 \sqrt [4]{3} a^{4/3} b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 108, normalized size = 0.36 \[ \frac {2 e^2 \sqrt {e x} \left (-7 a^2 B+\left (a+b x^3\right ) \sqrt {\frac {b x^3}{a}+1} (7 a B+2 A b) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\frac {b x^3}{a}\right )-2 a b \left (A+5 B x^3\right )+A b^2 x^3\right )}{27 a b^2 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{2} x^{5} + A e^{2} x^{2}\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.00, size = 7083, normalized size = 23.69 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^3+A\right )\,{\left (e\,x\right )}^{5/2}}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \]
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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