Optimal. Leaf size=144 \[ \frac {2 \sqrt {x} (a+b x) (A b-a B)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {a} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{3/2} (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \begin {gather*} \frac {2 \sqrt {x} (a+b x) (A b-a B)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {a} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{3/2} (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {\sqrt {x} (A+B x)}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B x^{3/2} (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (\frac {3 A b^2}{2}-\frac {3 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{a b+b^2 x} \, dx}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) \sqrt {x} (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{3/2} (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 a \left (\frac {3 A b^2}{2}-\frac {3 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}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) \sqrt {x} (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{3/2} (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (4 a \left (\frac {3 A b^2}{2}-\frac {3 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 )}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) \sqrt {x} (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{3/2} (a+b x)}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {a} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 0.57 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {x} (-3 a B+3 A b+b B x)+3 \sqrt {a} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{3 b^{5/2} \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.27, size = 84, normalized size = 0.58 \begin {gather*} \frac {(a+b x) \left (\frac {2 \left (a^{3/2} B-\sqrt {a} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {2 \sqrt {x} (-3 a B+3 A b+b B x)}{3 b^2}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 129, normalized size = 0.90 \begin {gather*} \left [-\frac {3 \, {\left (B a - A b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt {x}}{3 \, b^{2}}, \frac {2 \, {\left (3 \, {\left (B a - A b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt {x}\right )}}{3 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 94, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (B a^{2} \mathrm {sgn}\left (b x + a\right ) - A a b \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (B b^{2} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) - 3 \, B a b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 3 \, A b^{2} \sqrt {x} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 94, normalized size = 0.65 \begin {gather*} \frac {2 \left (b x +a \right ) \left (-3 A a b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3 B \,a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+\sqrt {a b}\, B b \,x^{\frac {3}{2}}+3 \sqrt {a b}\, A b \sqrt {x}-3 \sqrt {a b}\, B a \sqrt {x}\right )}{3 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {a b}\, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 122, normalized size = 0.85 \begin {gather*} -\frac {{\left ({\left (B a b - A b^{2}\right )} x^{2} + {\left (B a^{2} - A a b\right )} x\right )} \sqrt {x}}{a b^{2} x + a^{2} b} + \frac {2 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {{\left (5 \, B a b - 3 \, A b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (B a^{2} - A a b\right )} \sqrt {x}}{3 \, a b^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x}\,\left (A+B\,x\right )}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x} \left (A + B x\right )}{\sqrt {\left (a + b x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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