Optimal. Leaf size=238 \[ \frac {2 (a+b x) (A b-a B)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (a+b x) (A b-a B)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (a+b x) (A b-a B)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 78, 51, 63, 205} \[ \frac {2 b^2 (a+b x) (A b-a B)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (a+b x) (A b-a B)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (A b-a B)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {A+B x}{x^{9/2} \left (a b+b^2 x\right )} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{7/2} \left (a b+b^2 x\right )} \, dx}{7 a b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )} \, dx}{7 a^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 b \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{7 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 b^2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{7 a^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (4 b^2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{7 a^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 60, normalized size = 0.25 \[ -\frac {2 (a+b x) \left (\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\frac {b x}{a}\right ) (7 a B x-7 A b x)+5 a A\right )}{35 a^2 x^{7/2} \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 246, normalized size = 1.03 \[ \left [-\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}}{105 \, a^{4} x^{4}}, \frac {2 \, {\left (105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}\right )}}{105 \, a^{4} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 158, normalized size = 0.66 \[ -\frac {2 \, {\left (B a b^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {2 \, {\left (105 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) - 105 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) - 35 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) - 21 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, a^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 165, normalized size = 0.69 \[ \frac {2 \left (b x +a \right ) \left (105 A \,b^{4} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-105 B a \,b^{3} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 \sqrt {a b}\, A \,b^{3} x^{3}-105 \sqrt {a b}\, B a \,b^{2} x^{3}-35 \sqrt {a b}\, A a \,b^{2} x^{2}+35 \sqrt {a b}\, B \,a^{2} b \,x^{2}+21 \sqrt {a b}\, A \,a^{2} b x -21 \sqrt {a b}\, B \,a^{3} x -15 \sqrt {a b}\, A \,a^{3}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {a b}\, a^{4} x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.51, size = 357, normalized size = 1.50 \[ -\frac {35 \, {\left ({\left (5 \, B a b^{5} - 7 \, A b^{6}\right )} x^{2} + 3 \, {\left (7 \, B a^{2} b^{4} - 9 \, A a b^{5}\right )} x\right )} \sqrt {x} - \frac {70 \, {\left ({\left (5 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{2} - 3 \, {\left (7 \, B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x\right )}}{\sqrt {x}} - \frac {70 \, {\left (3 \, {\left (5 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{2} - {\left (7 \, B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x\right )}}{x^{\frac {3}{2}}} - \frac {14 \, {\left (5 \, {\left (5 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{2} + {\left (7 \, B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )}}{x^{\frac {5}{2}}} + \frac {2 \, {\left (7 \, {\left (5 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{6} - 9 \, A a^{5} b\right )} x\right )}}{x^{\frac {7}{2}}} + \frac {6 \, {\left (7 \, A a^{5} b x^{2} + 5 \, A a^{6} x\right )}}{x^{\frac {9}{2}}}}{105 \, {\left (a^{6} b x + a^{7}\right )}} - \frac {2 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {{\left (5 \, B a b^{4} - 7 \, A b^{5}\right )} x^{\frac {3}{2}} + 6 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} \sqrt {x}}{3 \, a^{6}} \]
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 \[ \int \frac {A+B\,x}{x^{9/2}\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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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