Optimal. Leaf size=611 \[ -\frac {A}{7 a x^7 \sqrt {a+b x^3}}-\frac {17 A b-14 a B}{21 a^2 x^4 \sqrt {a+b x^3}}+\frac {11 (17 A b-14 a B) \sqrt {a+b x^3}}{168 a^3 x^4}-\frac {55 b (17 A b-14 a B) \sqrt {a+b x^3}}{336 a^4 x}+\frac {55 b^{4/3} (17 A b-14 a B) \sqrt {a+b x^3}}{336 a^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {55 \sqrt {2-\sqrt {3}} b^{4/3} (17 A b-14 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224\ 3^{3/4} a^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {55 b^{4/3} (17 A b-14 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{168 \sqrt {2} \sqrt [4]{3} a^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.26, antiderivative size = 611, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {464, 296, 331,
309, 224, 1891} \begin {gather*} \frac {55 b^{4/3} \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}} (17 A b-14 a B) F\left (\text {ArcSin}\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 )}{168 \sqrt {2} \sqrt [4]{3} a^{11/3} \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 {55 \sqrt {2-\sqrt {3}} b^{4/3} \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}} (17 A b-14 a B) E\left (\text {ArcSin}\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 )}{224\ 3^{3/4} a^{11/3} \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 {55 b^{4/3} \sqrt {a+b x^3} (17 A b-14 a B)}{336 a^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {55 b \sqrt {a+b x^3} (17 A b-14 a B)}{336 a^4 x}+\frac {11 \sqrt {a+b x^3} (17 A b-14 a B)}{168 a^3 x^4}-\frac {17 A b-14 a B}{21 a^2 x^4 \sqrt {a+b x^3}}-\frac {A}{7 a x^7 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 296
Rule 309
Rule 331
Rule 464
Rule 1891
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^8 \left (a+b x^3\right )^{3/2}} \, dx &=-\frac {A}{7 a x^7 \sqrt {a+b x^3}}-\frac {\left (\frac {17 A b}{2}-7 a B\right ) \int \frac {1}{x^5 \left (a+b x^3\right )^{3/2}} \, dx}{7 a}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^3}}-\frac {17 A b-14 a B}{21 a^2 x^4 \sqrt {a+b x^3}}-\frac {(11 (17 A b-14 a B)) \int \frac {1}{x^5 \sqrt {a+b x^3}} \, dx}{42 a^2}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^3}}-\frac {17 A b-14 a B}{21 a^2 x^4 \sqrt {a+b x^3}}+\frac {11 (17 A b-14 a B) \sqrt {a+b x^3}}{168 a^3 x^4}+\frac {(55 b (17 A b-14 a B)) \int \frac {1}{x^2 \sqrt {a+b x^3}} \, dx}{336 a^3}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^3}}-\frac {17 A b-14 a B}{21 a^2 x^4 \sqrt {a+b x^3}}+\frac {11 (17 A b-14 a B) \sqrt {a+b x^3}}{168 a^3 x^4}-\frac {55 b (17 A b-14 a B) \sqrt {a+b x^3}}{336 a^4 x}+\frac {\left (55 b^2 (17 A b-14 a B)\right ) \int \frac {x}{\sqrt {a+b x^3}} \, dx}{672 a^4}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^3}}-\frac {17 A b-14 a B}{21 a^2 x^4 \sqrt {a+b x^3}}+\frac {11 (17 A b-14 a B) \sqrt {a+b x^3}}{168 a^3 x^4}-\frac {55 b (17 A b-14 a B) \sqrt {a+b x^3}}{336 a^4 x}+\frac {\left (55 b^{5/3} (17 A b-14 a B)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{672 a^4}+\frac {\left (55 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} b^{5/3} (17 A b-14 a B)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{336 a^{11/3}}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^3}}-\frac {17 A b-14 a B}{21 a^2 x^4 \sqrt {a+b x^3}}+\frac {11 (17 A b-14 a B) \sqrt {a+b x^3}}{168 a^3 x^4}-\frac {55 b (17 A b-14 a B) \sqrt {a+b x^3}}{336 a^4 x}+\frac {55 b^{4/3} (17 A b-14 a B) \sqrt {a+b x^3}}{336 a^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {55 \sqrt {2-\sqrt {3}} b^{4/3} (17 A b-14 a B) \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 )}{224\ 3^{3/4} a^{11/3} \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 {55 b^{4/3} (17 A b-14 a B) \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 )}{168 \sqrt {2} \sqrt [4]{3} a^{11/3} \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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 72, normalized size = 0.12 \begin {gather*} \frac {-8 a A+(17 A b-14 a B) x^3 \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (-\frac {4}{3},\frac {3}{2};-\frac {1}{3};-\frac {b x^3}{a}\right )}{56 a^2 x^7 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1017 vs. \(2 (465 ) = 930\).
time = 0.35, size = 1018, normalized size = 1.67
method | result | size |
elliptic | \(-\frac {A \sqrt {b \,x^{3}+a}}{7 a^{2} x^{7}}+\frac {\left (25 A b -14 B a \right ) \sqrt {b \,x^{3}+a}}{56 a^{3} x^{4}}-\frac {\left (237 A b -182 B a \right ) b \sqrt {b \,x^{3}+a}}{112 a^{4} x}-\frac {2 b^{2} x^{2} \left (A b -B a \right )}{3 a^{4} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 i \left (\frac {b^{2} \left (237 A b -182 B a \right )}{224 a^{4}}+\frac {b^{2} \left (A b -B a \right )}{3 a^{4}}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{3 b \sqrt {b \,x^{3}+a}}\) | \(573\) |
risch | \(\text {Expression too large to display}\) | \(1010\) |
default | \(\text {Expression too large to display}\) | \(1018\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.48, size = 156, normalized size = 0.26 \begin {gather*} \frac {55 \, {\left ({\left (14 \, B a b^{2} - 17 \, A b^{3}\right )} x^{10} + {\left (14 \, B a^{2} b - 17 \, A a b^{2}\right )} x^{7}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (55 \, {\left (14 \, B a b^{2} - 17 \, A b^{3}\right )} x^{9} + 33 \, {\left (14 \, B a^{2} b - 17 \, A a b^{2}\right )} x^{6} - 48 \, A a^{3} - 6 \, {\left (14 \, B a^{3} - 17 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{336 \, {\left (a^{4} b x^{10} + a^{5} x^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 30.16, size = 94, normalized size = 0.15 \begin {gather*} \frac {A \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, \frac {3}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {B \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {3}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x^{4} \Gamma \left (- \frac {1}{3}\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 {B\,x^3+A}{x^8\,{\left (b\,x^3+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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