Optimal. Leaf size=95 \[ \frac {(a+b x)^{3/2} (2 a B+3 A b)}{3 a}+\sqrt {a+b x} (2 a B+3 A b)-\sqrt {a} (2 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {A (a+b x)^{5/2}}{a x} \]
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Rubi [A] time = 0.04, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 50, 63, 208} \begin {gather*} \frac {(a+b x)^{3/2} (2 a B+3 A b)}{3 a}+\sqrt {a+b x} (2 a B+3 A b)-\sqrt {a} (2 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {A (a+b x)^{5/2}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^2} \, dx &=-\frac {A (a+b x)^{5/2}}{a x}+\frac {\left (\frac {3 A b}{2}+a B\right ) \int \frac {(a+b x)^{3/2}}{x} \, dx}{a}\\ &=\frac {(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac {A (a+b x)^{5/2}}{a x}+\frac {1}{2} (3 A b+2 a B) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=(3 A b+2 a B) \sqrt {a+b x}+\frac {(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac {A (a+b x)^{5/2}}{a x}+\frac {1}{2} (a (3 A b+2 a B)) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=(3 A b+2 a B) \sqrt {a+b x}+\frac {(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac {A (a+b x)^{5/2}}{a x}+\frac {(a (3 A b+2 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=(3 A b+2 a B) \sqrt {a+b x}+\frac {(3 A b+2 a B) (a+b x)^{3/2}}{3 a}-\frac {A (a+b x)^{5/2}}{a x}-\sqrt {a} (3 A b+2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 71, normalized size = 0.75 \begin {gather*} \frac {\sqrt {a+b x} (a (8 B x-3 A)+2 b x (3 A+B x))}{3 x}-\sqrt {a} (2 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 95, normalized size = 1.00 \begin {gather*} \left (-2 a^{3/2} B-3 \sqrt {a} A b\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {\sqrt {a+b x} \left (-6 a^2 B+6 A b (a+b x)-9 a A b+4 a B (a+b x)+2 B (a+b x)^2\right )}{3 b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.69, size = 151, normalized size = 1.59 \begin {gather*} \left [\frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {a} x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, B b x^{2} - 3 \, A a + 2 \, {\left (4 \, B a + 3 \, A b\right )} x\right )} \sqrt {b x + a}}{6 \, x}, \frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (2 \, B b x^{2} - 3 \, A a + 2 \, {\left (4 \, B a + 3 \, A b\right )} x\right )} \sqrt {b x + a}}{3 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 93, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} B b + 6 \, \sqrt {b x + a} B a b + 6 \, \sqrt {b x + a} A b^{2} - \frac {3 \, \sqrt {b x + a} A a b}{x} + \frac {3 \, {\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 77, normalized size = 0.81 \begin {gather*} 2 \sqrt {b x +a}\, A b +2 \sqrt {b x +a}\, B a +\frac {2 \left (b x +a \right )^{\frac {3}{2}} B}{3}+2 \left (-\frac {\left (3 A b +2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {b x +a}\, A}{2 x}\right ) a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 97, normalized size = 1.02 \begin {gather*} \frac {1}{6} \, {\left (\frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {a} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{b} - \frac {6 \, \sqrt {b x + a} A a}{b x} + \frac {4 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} B + 3 \, {\left (B a + A b\right )} \sqrt {b x + a}\right )}}{b}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 96, normalized size = 1.01 \begin {gather*} \left (2\,A\,b+2\,B\,a\right )\,\sqrt {a+b\,x}+\frac {2\,B\,{\left (a+b\,x\right )}^{3/2}}{3}+2\,\mathrm {atan}\left (\frac {2\,\left (3\,A\,b+2\,B\,a\right )\,\sqrt {-\frac {a}{4}}\,\sqrt {a+b\,x}}{2\,B\,a^2+3\,A\,b\,a}\right )\,\left (3\,A\,b+2\,B\,a\right )\,\sqrt {-\frac {a}{4}}-\frac {A\,a\,\sqrt {a+b\,x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.56, size = 202, normalized size = 2.13 \begin {gather*} - \frac {A a^{2} b \sqrt {\frac {1}{a^{3}}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {A a^{2} b \sqrt {\frac {1}{a^{3}}} \log {\left (a^{2} \sqrt {\frac {1}{a^{3}}} + \sqrt {a + b x} \right )}}{2} + \frac {4 A a b \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} - \frac {A a \sqrt {a + b x}}{x} + 2 A b \sqrt {a + b x} + \frac {2 B a^{2} \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 B a \sqrt {a + b x} + B b \left (\begin {cases} \sqrt {a} x & \text {for}\: b = 0 \\\frac {2 \left (a + b x\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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