Optimal. Leaf size=117 \[ \frac {b (2 A b+3 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\sqrt {b} (2 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {79, 49, 52, 65,
223, 212} \begin {gather*} -\frac {2 (a+b x)^{3/2} (3 a B+2 A b)}{3 a \sqrt {x}}+\frac {b \sqrt {x} \sqrt {a+b x} (3 a B+2 A b)}{a}+\sqrt {b} (3 a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^{5/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\frac {\left (2 \left (A b+\frac {3 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx}{3 a}\\ &=-\frac {2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\frac {(b (2 A b+3 a B)) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx}{a}\\ &=\frac {b (2 A b+3 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\frac {1}{2} (b (2 A b+3 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {b (2 A b+3 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}}+(b (2 A b+3 a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {b (2 A b+3 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}}+(b (2 A b+3 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {b (2 A b+3 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 (2 A b+3 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{5/2}}{3 a x^{3/2}}+\sqrt {b} (2 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 80, normalized size = 0.68 \begin {gather*} \frac {\sqrt {a+b x} \left (-2 a A-8 A b x-6 a B x+3 b B x^2\right )}{3 x^{3/2}}-\sqrt {b} (2 A b+3 a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 162, normalized size = 1.38
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-3 b B \,x^{2}+8 A b x +6 B a x +2 A a \right )}{3 x^{\frac {3}{2}}}+\frac {\left (A \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) b^{\frac {3}{2}}+\frac {3 B \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {b}\, a}{2}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\) | \(117\) |
default | \(\frac {\sqrt {b x +a}\, \left (6 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) b^{2} x^{2}+9 B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a b \,x^{2}+6 B \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, x^{2}-16 A \,b^{\frac {3}{2}} x \sqrt {\left (b x +a \right ) x}-12 B a x \sqrt {b}\, \sqrt {\left (b x +a \right ) x}-4 A a \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\right )}{6 x^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}\) | \(162\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 146, normalized size = 1.25 \begin {gather*} \frac {3}{2} \, B a \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + A b^{\frac {3}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {3 \, \sqrt {b x^{2} + a x} B a}{x} - \frac {7 \, \sqrt {b x^{2} + a x} A b}{3 \, x} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{x^{2}} - \frac {\sqrt {b x^{2} + a x} A a}{3 \, x^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 163, normalized size = 1.39 \begin {gather*} \left [\frac {3 \, {\left (3 \, B a + 2 \, A b\right )} \sqrt {b} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (3 \, B b x^{2} - 2 \, A a - 2 \, {\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{6 \, x^{2}}, -\frac {3 \, {\left (3 \, B a + 2 \, A b\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (3 \, B b x^{2} - 2 \, A a - 2 \, {\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 19.02, size = 168, normalized size = 1.44 \begin {gather*} A \left (- \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {8 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3} - b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )} + 2 b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x} + 1} + 1 \right )}\right ) + B \left (- \frac {2 a^{\frac {3}{2}}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} - \frac {\sqrt {a} b \sqrt {x}}{\sqrt {1 + \frac {b x}{a}}} + 3 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + \frac {b^{2} x^{\frac {3}{2}}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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 {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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