Optimal. Leaf size=113 \[ \frac {2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {2 b^2 (A b-a B)}{a^4 \sqrt {x}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 A}{7 a x^{7/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 51, 63, 205} \[ \frac {2 b^2 (A b-a B)}{a^4 \sqrt {x}}+\frac {2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 A}{7 a x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{9/2} (a+b x)} \, dx &=-\frac {2 A}{7 a x^{7/2}}+\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {7 a B}{2}\right )\right ) \int \frac {1}{x^{7/2} (a+b x)} \, dx}{7 a}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}+\frac {(b (A b-a B)) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{a^2}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}-\frac {\left (b^2 (A b-a B)\right ) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{a^3}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac {2 b^2 (A b-a B)}{a^4 \sqrt {x}}+\frac {\left (b^3 (A b-a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{a^4}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac {2 b^2 (A b-a B)}{a^4 \sqrt {x}}+\frac {\left (2 b^3 (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^4}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac {2 b^2 (A b-a B)}{a^4 \sqrt {x}}+\frac {2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 44, normalized size = 0.39 \[ -\frac {2 \left (\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\frac {b x}{a}\right ) (7 a B x-7 A b x)+5 a A\right )}{35 a^2 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 246, normalized size = 2.18 \[ \left [-\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}}{105 \, a^{4} x^{4}}, \frac {2 \, {\left (105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}\right )}}{105 \, a^{4} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 104, normalized size = 0.92 \[ -\frac {2 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {2 \, {\left (105 \, B a b^{2} x^{3} - 105 \, A b^{3} x^{3} - 35 \, B a^{2} b x^{2} + 35 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 21 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 126, normalized size = 1.12 \[ \frac {2 A \,b^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}-\frac {2 B \,b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {2 A \,b^{3}}{a^{4} \sqrt {x}}-\frac {2 B \,b^{2}}{a^{3} \sqrt {x}}-\frac {2 A \,b^{2}}{3 a^{3} x^{\frac {3}{2}}}+\frac {2 B b}{3 a^{2} x^{\frac {3}{2}}}+\frac {2 A b}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 B}{5 a \,x^{\frac {5}{2}}}-\frac {2 A}{7 a \,x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 103, normalized size = 0.91 \[ -\frac {2 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {2 \, {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )}}{105 \, a^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 90, normalized size = 0.80 \[ \frac {2\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{9/2}}-\frac {\frac {2\,A}{7\,a}-\frac {2\,x\,\left (A\,b-B\,a\right )}{5\,a^2}-\frac {2\,b^2\,x^3\,\left (A\,b-B\,a\right )}{a^4}+\frac {2\,b\,x^2\,\left (A\,b-B\,a\right )}{3\,a^3}}{x^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 79.17, size = 326, normalized size = 2.88 \[ \begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{a} & \text {for}\: b = 0 \\- \frac {2 A}{7 a x^{\frac {7}{2}}} + \frac {2 A b}{5 a^{2} x^{\frac {5}{2}}} - \frac {2 A b^{2}}{3 a^{3} x^{\frac {3}{2}}} + \frac {2 A b^{3}}{a^{4} \sqrt {x}} - \frac {i A b^{3} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {9}{2}} \sqrt {\frac {1}{b}}} + \frac {i A b^{3} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {9}{2}} \sqrt {\frac {1}{b}}} - \frac {2 B}{5 a x^{\frac {5}{2}}} + \frac {2 B b}{3 a^{2} x^{\frac {3}{2}}} - \frac {2 B b^{2}}{a^{3} \sqrt {x}} + \frac {i B b^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {7}{2}} \sqrt {\frac {1}{b}}} - \frac {i B b^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {7}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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