Optimal. Leaf size=577 \[ -\frac {2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac {20 (7 A b-16 a B) x^2}{189 b^3 \sqrt {a+b x^3}}+\frac {80 (7 A b-16 a B) \sqrt {a+b x^3}}{189 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {40 \sqrt {2-\sqrt {3}} \sqrt [3]{a} (7 A b-16 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{63\ 3^{3/4} b^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {80 \sqrt {2} \sqrt [3]{a} (7 A b-16 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{189 \sqrt [4]{3} b^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.23, antiderivative size = 577, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {470, 294, 309,
224, 1891} \begin {gather*} \frac {80 \sqrt {2} \sqrt [3]{a} \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}} (7 A b-16 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 )}{189 \sqrt [4]{3} b^{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 {40 \sqrt {2-\sqrt {3}} \sqrt [3]{a} \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}} (7 A b-16 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 )}{63\ 3^{3/4} b^{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 {80 \sqrt {a+b x^3} (7 A b-16 a B)}{189 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {20 x^2 (7 A b-16 a B)}{189 b^3 \sqrt {a+b x^3}}-\frac {2 x^5 (7 A b-16 a B)}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 294
Rule 309
Rule 470
Rule 1891
Rubi steps
\begin {align*} \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac {\left (2 \left (-\frac {7 A b}{2}+8 a B\right )\right ) \int \frac {x^7}{\left (a+b x^3\right )^{5/2}} \, dx}{7 b}\\ &=-\frac {2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}+\frac {(10 (7 A b-16 a B)) \int \frac {x^4}{\left (a+b x^3\right )^{3/2}} \, dx}{63 b^2}\\ &=-\frac {2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac {20 (7 A b-16 a B) x^2}{189 b^3 \sqrt {a+b x^3}}+\frac {(40 (7 A b-16 a B)) \int \frac {x}{\sqrt {a+b x^3}} \, dx}{189 b^3}\\ &=-\frac {2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac {20 (7 A b-16 a B) x^2}{189 b^3 \sqrt {a+b x^3}}+\frac {(40 (7 A b-16 a B)) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{189 b^{10/3}}+\frac {\left (40 \sqrt {2 \left (2-\sqrt {3}\right )} \sqrt [3]{a} (7 A b-16 a B)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{189 b^{10/3}}\\ &=-\frac {2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac {20 (7 A b-16 a B) x^2}{189 b^3 \sqrt {a+b x^3}}+\frac {80 (7 A b-16 a B) \sqrt {a+b x^3}}{189 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {40 \sqrt {2-\sqrt {3}} \sqrt [3]{a} (7 A b-16 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 )}{63\ 3^{3/4} b^{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 {80 \sqrt {2} \sqrt [3]{a} (7 A b-16 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 )}{189 \sqrt [4]{3} b^{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 = 10.10, size = 109, normalized size = 0.19 \begin {gather*} \frac {2 x^2 \left (-32 a^2 B+2 a b \left (7 A-8 B x^3\right )+b^2 x^3 \left (7 A+B x^3\right )+2 (-7 A b+16 a B) \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (\frac {2}{3},\frac {5}{2};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{7 b^3 \left (a+b x^3\right )^{3/2}} \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. 996 vs. \(2 (435 ) = 870\).
time = 0.38, size = 997, normalized size = 1.73
method | result | size |
elliptic | \(\frac {2 a \,x^{2} \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 b^{5} \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {2 x^{2} \left (13 A b -22 B a \right )}{27 b^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 B \,x^{2} \sqrt {b \,x^{3}+a}}{7 b^{3}}-\frac {2 i \left (\frac {A b -2 B a}{b^{3}}+\frac {13 A b -22 B a}{27 b^{3}}-\frac {4 B a}{7 b^{3}}\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}}\) | \(555\) |
default | \(\text {Expression too large to display}\) | \(997\) |
risch | \(\text {Expression too large to display}\) | \(1443\) |
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 = 0.58, size = 162, normalized size = 0.28 \begin {gather*} \frac {2 \, {\left (40 \, {\left ({\left (16 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 16 \, B a^{3} - 7 \, A a^{2} b + 2 \, {\left (16 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (27 \, B b^{3} x^{8} + 13 \, {\left (16 \, B a b^{2} - 7 \, A b^{3}\right )} x^{5} + 10 \, {\left (16 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{189 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 53.89, size = 80, normalized size = 0.14 \begin {gather*} \frac {A x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {11}{3}\right )} + \frac {B x^{11} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {14}{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 {x^7\,\left (B\,x^3+A\right )}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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