Optimal. Leaf size=136 \[ \frac {2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}-\frac {2 a^3 \sqrt {x} (A b-a B)}{b^5}+\frac {2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac {2 a x^{5/2} (A b-a B)}{5 b^3}+\frac {2 x^{7/2} (A b-a B)}{7 b^2}+\frac {2 B x^{9/2}}{9 b} \]
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Rubi [A] time = 0.08, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 63, 205} \[ \frac {2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac {2 a^3 \sqrt {x} (A b-a B)}{b^5}+\frac {2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}+\frac {2 x^{7/2} (A b-a B)}{7 b^2}-\frac {2 a x^{5/2} (A b-a B)}{5 b^3}+\frac {2 B x^{9/2}}{9 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 205
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{a+b x} \, dx &=\frac {2 B x^{9/2}}{9 b}+\frac {\left (2 \left (\frac {9 A b}{2}-\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{a+b x} \, dx}{9 b}\\ &=\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {(a (A b-a B)) \int \frac {x^{5/2}}{a+b x} \, dx}{b^2}\\ &=-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {x^{3/2}}{a+b x} \, dx}{b^3}\\ &=\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {\left (a^3 (A b-a B)\right ) \int \frac {\sqrt {x}}{a+b x} \, dx}{b^4}\\ &=-\frac {2 a^3 (A b-a B) \sqrt {x}}{b^5}+\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (a^4 (A b-a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^5}\\ &=-\frac {2 a^3 (A b-a B) \sqrt {x}}{b^5}+\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {\left (2 a^4 (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^5}\\ &=-\frac {2 a^3 (A b-a B) \sqrt {x}}{b^5}+\frac {2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac {2 a (A b-a B) x^{5/2}}{5 b^3}+\frac {2 (A b-a B) x^{7/2}}{7 b^2}+\frac {2 B x^{9/2}}{9 b}+\frac {2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 120, normalized size = 0.88 \[ \frac {2 \sqrt {x} \left (315 a^4 B-105 a^3 b (3 A+B x)+21 a^2 b^2 x (5 A+3 B x)-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )}{315 b^5}-\frac {2 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 276, normalized size = 2.03 \[ \left [-\frac {315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{315 \, b^{5}}, -\frac {2 \, {\left (315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt {x}\right )}}{315 \, b^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 139, normalized size = 1.02 \[ -\frac {2 \, {\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (35 \, B b^{8} x^{\frac {9}{2}} - 45 \, B a b^{7} x^{\frac {7}{2}} + 45 \, A b^{8} x^{\frac {7}{2}} + 63 \, B a^{2} b^{6} x^{\frac {5}{2}} - 63 \, A a b^{7} x^{\frac {5}{2}} - 105 \, B a^{3} b^{5} x^{\frac {3}{2}} + 105 \, A a^{2} b^{6} x^{\frac {3}{2}} + 315 \, B a^{4} b^{4} \sqrt {x} - 315 \, A a^{3} b^{5} \sqrt {x}\right )}}{315 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 150, normalized size = 1.10 \[ \frac {2 B \,x^{\frac {9}{2}}}{9 b}+\frac {2 A \,x^{\frac {7}{2}}}{7 b}-\frac {2 B a \,x^{\frac {7}{2}}}{7 b^{2}}-\frac {2 A a \,x^{\frac {5}{2}}}{5 b^{2}}+\frac {2 B \,a^{2} x^{\frac {5}{2}}}{5 b^{3}}+\frac {2 A \,a^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {2 B \,a^{5} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{5}}+\frac {2 A \,a^{2} x^{\frac {3}{2}}}{3 b^{3}}-\frac {2 B \,a^{3} x^{\frac {3}{2}}}{3 b^{4}}-\frac {2 A \,a^{3} \sqrt {x}}{b^{4}}+\frac {2 B \,a^{4} \sqrt {x}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.89, size = 128, normalized size = 0.94 \[ -\frac {2 \, {\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (35 \, B b^{4} x^{\frac {9}{2}} - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{\frac {7}{2}} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{\frac {5}{2}} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{\frac {3}{2}} + 315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {x}\right )}}{315 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 151, normalized size = 1.11 \[ x^{7/2}\,\left (\frac {2\,A}{7\,b}-\frac {2\,B\,a}{7\,b^2}\right )+\frac {2\,B\,x^{9/2}}{9\,b}+\frac {a^2\,x^{3/2}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{3\,b^2}-\frac {a^3\,\sqrt {x}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{b^3}-\frac {2\,a^{7/2}\,\mathrm {atan}\left (\frac {a^{7/2}\,\sqrt {b}\,\sqrt {x}\,\left (A\,b-B\,a\right )}{B\,a^5-A\,a^4\,b}\right )\,\left (A\,b-B\,a\right )}{b^{11/2}}-\frac {a\,x^{5/2}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{5\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 52.58, size = 313, normalized size = 2.30 \[ \begin {cases} - \frac {i A a^{\frac {7}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{5} \sqrt {\frac {1}{b}}} + \frac {i A a^{\frac {7}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{5} \sqrt {\frac {1}{b}}} - \frac {2 A a^{3} \sqrt {x}}{b^{4}} + \frac {2 A a^{2} x^{\frac {3}{2}}}{3 b^{3}} - \frac {2 A a x^{\frac {5}{2}}}{5 b^{2}} + \frac {2 A x^{\frac {7}{2}}}{7 b} + \frac {i B a^{\frac {9}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{6} \sqrt {\frac {1}{b}}} - \frac {i B a^{\frac {9}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{6} \sqrt {\frac {1}{b}}} + \frac {2 B a^{4} \sqrt {x}}{b^{5}} - \frac {2 B a^{3} x^{\frac {3}{2}}}{3 b^{4}} + \frac {2 B a^{2} x^{\frac {5}{2}}}{5 b^{3}} - \frac {2 B a x^{\frac {7}{2}}}{7 b^{2}} + \frac {2 B x^{\frac {9}{2}}}{9 b} & \text {for}\: b \neq 0 \\\frac {\frac {2 A x^{\frac {9}{2}}}{9} + \frac {2 B x^{\frac {11}{2}}}{11}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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