Optimal. Leaf size=190 \[ \frac {2 (a+b x) (A b-a B)}{3 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A (a+b x)}{5 a x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (a+b x) (A b-a B)}{a^3 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {2 b (a+b x) (A b-a B)}{a^3 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (A b-a B)}{3 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A (a+b x)}{5 a x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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^{7/2} \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^{7/2} \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)}{5 a x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (-\frac {5 A b^2}{2}+\frac {5 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}{5 a b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{5 a x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{3 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 \left (-\frac {5 A b^2}{2}+\frac {5 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}{5 a^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{5 a x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{3 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{a^3 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 b \left (-\frac {5 A b^2}{2}+\frac {5 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}{5 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{5 a x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{3 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{a^3 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (4 b \left (-\frac {5 A b^2}{2}+\frac {5 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 )}{5 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A (a+b x)}{5 a x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{3 a^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{a^3 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b^{3/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2} \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 = 0.32 \[ -\frac {2 (a+b x) \left (\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b x}{a}\right ) (5 a B x-5 A b x)+3 a A\right )}{15 a^2 x^{5/2} \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 195, normalized size = 1.03 \[ \left [-\frac {15 \, {\left (B a b - A b^{2}\right )} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (3 \, A a^{2} - 15 \, {\left (B a b - A b^{2}\right )} x^{2} + 5 \, {\left (B a^{2} - A a b\right )} x\right )} \sqrt {x}}{15 \, a^{3} x^{3}}, -\frac {2 \, {\left (15 \, {\left (B a b - A b^{2}\right )} x^{3} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (3 \, A a^{2} - 15 \, {\left (B a b - A b^{2}\right )} x^{2} + 5 \, {\left (B a^{2} - A a b\right )} x\right )} \sqrt {x}\right )}}{15 \, a^{3} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 122, normalized size = 0.64 \[ \frac {2 \, {\left (B a b^{2} \mathrm {sgn}\left (b x + a\right ) - A b^{3} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {2 \, {\left (15 \, B a b x^{2} \mathrm {sgn}\left (b x + a\right ) - 15 \, A b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) - 5 \, B a^{2} x \mathrm {sgn}\left (b x + a\right ) + 5 \, A a b x \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} \mathrm {sgn}\left (b x + a\right )\right )}}{15 \, a^{3} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 131, normalized size = 0.69 \[ -\frac {2 \left (b x +a \right ) \left (15 A \,b^{3} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-15 B a \,b^{2} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+15 \sqrt {a b}\, A \,b^{2} x^{2}-15 \sqrt {a b}\, B a b \,x^{2}-5 \sqrt {a b}\, A a b x +5 \sqrt {a b}\, B \,a^{2} x +3 \sqrt {a b}\, A \,a^{2}\right )}{15 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {a b}\, a^{3} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.65, size = 305, normalized size = 1.61 \[ \frac {5 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{2} + 3 \, {\left (5 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x\right )} \sqrt {x} - \frac {10 \, {\left ({\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{2} - 3 \, {\left (5 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x\right )}}{\sqrt {x}} - \frac {10 \, {\left (3 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{2} - {\left (5 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )}}{x^{\frac {3}{2}}} - \frac {2 \, {\left (5 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (5 \, B a^{5} - 7 \, A a^{4} b\right )} x\right )}}{x^{\frac {5}{2}}} - \frac {2 \, {\left (5 \, A a^{4} b x^{2} + 3 \, A a^{5} x\right )}}{x^{\frac {7}{2}}}}{15 \, {\left (a^{5} b x + a^{6}\right )}} + \frac {2 \, {\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{\frac {3}{2}} + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} \sqrt {x}}{3 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,x}{x^{7/2}\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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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