Optimal. Leaf size=564 \[ -\frac {2 (2 A b+7 a B) \sqrt {a+b x^3}}{7 a \sqrt {x}}+\frac {3 \left (1+\sqrt {3}\right ) \sqrt [3]{b} (2 A b+7 a B) \sqrt {x} \sqrt {a+b x^3}}{7 a \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}-\frac {3 \sqrt [4]{3} \sqrt [3]{b} (2 A b+7 a B) \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [3]{b} (2 A b+7 a B) \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{14 a^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.39, antiderivative size = 564, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {464, 283, 335,
314, 231, 1895} \begin {gather*} -\frac {3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt {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 (\text {ArcCos}\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 )}{14 a^{2/3} \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 {3 \sqrt [4]{3} \sqrt [3]{b} \sqrt {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) E\left (\text {ArcCos}\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 )}{7 a^{2/3} \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 \sqrt {a+b x^3} (7 a B+2 A b)}{7 a \sqrt {x}}+\frac {3 \left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt {x} \sqrt {a+b x^3} (7 a B+2 A b)}{7 a \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 231
Rule 283
Rule 314
Rule 335
Rule 464
Rule 1895
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx &=-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}-\frac {\left (2 \left (-A b-\frac {7 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x^3}}{x^{3/2}} \, dx}{7 a}\\ &=-\frac {2 (2 A b+7 a B) \sqrt {a+b x^3}}{7 a \sqrt {x}}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}+\frac {(3 b (2 A b+7 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x^3}} \, dx}{7 a}\\ &=-\frac {2 (2 A b+7 a B) \sqrt {a+b x^3}}{7 a \sqrt {x}}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}+\frac {(6 b (2 A b+7 a B)) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{7 a}\\ &=-\frac {2 (2 A b+7 a B) \sqrt {a+b x^3}}{7 a \sqrt {x}}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}-\frac {\left (3 \sqrt [3]{b} (2 A b+7 a B)\right ) \text {Subst}\left (\int \frac {\left (-1+\sqrt {3}\right ) a^{2/3}-2 b^{2/3} x^4}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{7 a}-\frac {\left (3 \left (1-\sqrt {3}\right ) \sqrt [3]{b} (2 A b+7 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{7 \sqrt [3]{a}}\\ &=-\frac {2 (2 A b+7 a B) \sqrt {a+b x^3}}{7 a \sqrt {x}}+\frac {3 \left (1+\sqrt {3}\right ) \sqrt [3]{b} (2 A b+7 a B) \sqrt {x} \sqrt {a+b x^3}}{7 a \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}-\frac {3 \sqrt [4]{3} \sqrt [3]{b} (2 A b+7 a B) \sqrt {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}} E\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 )}{7 a^{2/3} \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 {3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [3]{b} (2 A b+7 a B) \sqrt {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 )}{14 a^{2/3} \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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.07, size = 81, normalized size = 0.14 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (-A \left (a+b x^3\right )-\frac {(2 A b+7 a B) x^3 \, _2F_1\left (-\frac {1}{2},-\frac {1}{6};\frac {5}{6};-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{7 a x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.62, size = 5911, normalized size = 10.48
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1127\) |
elliptic | \(\text {Expression too large to display}\) | \(1177\) |
default | \(\text {Expression too large to display}\) | \(5911\) |
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 11.93, size = 97, normalized size = 0.17 \begin {gather*} \frac {A \sqrt {a} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {1}{2} \\ - \frac {1}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{\frac {7}{2}} \Gamma \left (- \frac {1}{6}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {x} \Gamma \left (\frac {5}{6}\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 {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{x^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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