Optimal. Leaf size=82 \[ \frac {b (A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {\sqrt {a+b x} (A b-4 a B)}{4 a x}-\frac {A (a+b x)^{3/2}}{2 a x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 47, 63, 208} \[ \frac {b (A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {\sqrt {a+b x} (A b-4 a B)}{4 a x}-\frac {A (a+b x)^{3/2}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x^3} \, dx &=-\frac {A (a+b x)^{3/2}}{2 a x^2}+\frac {\left (-\frac {A b}{2}+2 a B\right ) \int \frac {\sqrt {a+b x}}{x^2} \, dx}{2 a}\\ &=\frac {(A b-4 a B) \sqrt {a+b x}}{4 a x}-\frac {A (a+b x)^{3/2}}{2 a x^2}-\frac {(b (A b-4 a B)) \int \frac {1}{x \sqrt {a+b x}} \, dx}{8 a}\\ &=\frac {(A b-4 a B) \sqrt {a+b x}}{4 a x}-\frac {A (a+b x)^{3/2}}{2 a x^2}-\frac {(A b-4 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{4 a}\\ &=\frac {(A b-4 a B) \sqrt {a+b x}}{4 a x}-\frac {A (a+b x)^{3/2}}{2 a x^2}+\frac {b (A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 81, normalized size = 0.99 \[ \frac {-b x^2 \sqrt {\frac {b x}{a}+1} (4 a B-A b) \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )-(a+b x) (2 a (A+2 B x)+A b x)}{4 a x^2 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 158, normalized size = 1.93 \[ \left [-\frac {{\left (4 \, B a b - A b^{2}\right )} \sqrt {a} x^{2} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, A a^{2} + {\left (4 \, B a^{2} + A a b\right )} x\right )} \sqrt {b x + a}}{8 \, a^{2} x^{2}}, \frac {{\left (4 \, B a b - A b^{2}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (2 \, A a^{2} + {\left (4 \, B a^{2} + A a b\right )} x\right )} \sqrt {b x + a}}{4 \, a^{2} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 110, normalized size = 1.34 \[ \frac {\frac {{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b^{2} - 4 \, \sqrt {b x + a} B a^{2} b^{2} + {\left (b x + a\right )}^{\frac {3}{2}} A b^{3} + \sqrt {b x + a} A a b^{3}}{a b^{2} x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.91 \[ 2 \left (\frac {\left (A b -4 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}+\frac {-\frac {\left (A b +4 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{8 a}+\left (-\frac {A b}{8}+\frac {B a}{2}\right ) \sqrt {b x +a}}{b^{2} x^{2}}\right ) b \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.96, size = 120, normalized size = 1.46 \[ -\frac {1}{8} \, b^{2} {\left (\frac {2 \, {\left ({\left (4 \, B a + A b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - {\left (4 \, B a^{2} - A a b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{2} a b - 2 \, {\left (b x + a\right )} a^{2} b + a^{3} b} - \frac {{\left (4 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 94, normalized size = 1.15 \[ \frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-4\,B\,a\right )}{4\,a^{3/2}}-\frac {\left (\frac {A\,b^2}{4}-B\,a\,b\right )\,\sqrt {a+b\,x}+\frac {\left (A\,b^2+4\,B\,a\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{4\,a}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 36.42, size = 372, normalized size = 4.54 \[ - \frac {10 A a^{2} b^{2} \sqrt {a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac {6 A a b^{2} \left (a + b x\right )^{\frac {3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac {3 A a b^{2} \sqrt {\frac {1}{a^{5}}} \log {\left (- a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - \frac {3 A a b^{2} \sqrt {\frac {1}{a^{5}}} \log {\left (a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {a + b x} \right )}}{8} - \frac {A b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {A b^{2} \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} - \frac {A b \sqrt {a + b x}}{a x} - \frac {B a b \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {B a b \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {2 B b \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} - \frac {B \sqrt {a + b x}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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