Optimal. Leaf size=153 \[ \frac {b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{11/2}}+\frac {b^2 (9 A b-7 a B)}{a^5 \sqrt {x}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {A b-a B}{a b x^{7/2} (a+b x)} \]
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Rubi [A] time = 0.07, antiderivative size = 153, 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 = {78, 51, 63, 205} \[ \frac {b^2 (9 A b-7 a B)}{a^5 \sqrt {x}}+\frac {b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{11/2}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {A b-a B}{a b x^{7/2} (a+b x)} \]
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)^2} \, dx &=\frac {A b-a B}{a b x^{7/2} (a+b x)}-\frac {\left (-\frac {9 A b}{2}+\frac {7 a B}{2}\right ) \int \frac {1}{x^{9/2} (a+b x)} \, dx}{a b}\\ &=-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {A b-a B}{a b x^{7/2} (a+b x)}-\frac {(9 A b-7 a B) \int \frac {1}{x^{7/2} (a+b x)} \, dx}{2 a^2}\\ &=-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}+\frac {A b-a B}{a b x^{7/2} (a+b x)}+\frac {(b (9 A b-7 a B)) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{2 a^3}\\ &=-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {A b-a B}{a b x^{7/2} (a+b x)}-\frac {\left (b^2 (9 A b-7 a B)\right ) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a^4}\\ &=-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {b^2 (9 A b-7 a B)}{a^5 \sqrt {x}}+\frac {A b-a B}{a b x^{7/2} (a+b x)}+\frac {\left (b^3 (9 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^5}\\ &=-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {b^2 (9 A b-7 a B)}{a^5 \sqrt {x}}+\frac {A b-a B}{a b x^{7/2} (a+b x)}+\frac {\left (b^3 (9 A b-7 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^5}\\ &=-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {b^2 (9 A b-7 a B)}{a^5 \sqrt {x}}+\frac {A b-a B}{a b x^{7/2} (a+b x)}+\frac {b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 64, normalized size = 0.42 \[ \frac {(a+b x) (7 a B-9 A b) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\frac {b x}{a}\right )+7 a (A b-a B)}{7 a^2 b x^{7/2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 372, normalized size = 2.43 \[ \left [-\frac {105 \, {\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (30 \, A a^{4} + 105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{210 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}, \frac {105 \, {\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (30 \, A a^{4} + 105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{105 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 136, normalized size = 0.89 \[ -\frac {{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {B a b^{3} \sqrt {x} - A b^{4} \sqrt {x}}{{\left (b x + a\right )} a^{5}} - \frac {2 \, {\left (315 \, B a b^{2} x^{3} - 420 \, A b^{3} x^{3} - 70 \, B a^{2} b x^{2} + 105 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 42 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{5} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 163, normalized size = 1.07 \[ \frac {9 A \,b^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{5}}-\frac {7 B \,b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}+\frac {A \,b^{4} \sqrt {x}}{\left (b x +a \right ) a^{5}}-\frac {B \,b^{3} \sqrt {x}}{\left (b x +a \right ) a^{4}}+\frac {8 A \,b^{3}}{a^{5} \sqrt {x}}-\frac {6 B \,b^{2}}{a^{4} \sqrt {x}}-\frac {2 A \,b^{2}}{a^{4} x^{\frac {3}{2}}}+\frac {4 B b}{3 a^{3} x^{\frac {3}{2}}}+\frac {4 A b}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 B}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 A}{7 a^{2} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 143, normalized size = 0.93 \[ -\frac {30 \, A a^{4} + 105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x}{105 \, {\left (a^{5} b x^{\frac {9}{2}} + a^{6} x^{\frac {7}{2}}\right )}} - \frac {{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 121, normalized size = 0.79 \[ \frac {\frac {2\,x\,\left (9\,A\,b-7\,B\,a\right )}{35\,a^2}-\frac {2\,A}{7\,a}+\frac {2\,b^2\,x^3\,\left (9\,A\,b-7\,B\,a\right )}{3\,a^4}+\frac {b^3\,x^4\,\left (9\,A\,b-7\,B\,a\right )}{a^5}-\frac {2\,b\,x^2\,\left (9\,A\,b-7\,B\,a\right )}{15\,a^3}}{a\,x^{7/2}+b\,x^{9/2}}+\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (9\,A\,b-7\,B\,a\right )}{a^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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