Optimal. Leaf size=115 \[ \frac {3}{4} (4 A b+a B) \sqrt {x} \sqrt {a+b x}+\frac {(4 A b+a B) \sqrt {x} (a+b x)^{3/2}}{2 a}-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}}+\frac {3 a (4 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}} \]
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Rubi [A]
time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 52, 65, 223,
212} \begin {gather*} \frac {\sqrt {x} (a+b x)^{3/2} (a B+4 A b)}{2 a}+\frac {3}{4} \sqrt {x} \sqrt {a+b x} (a B+4 A b)+\frac {3 a (a B+4 A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}}-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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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^{3/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}}+\frac {\left (2 \left (2 A b+\frac {a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx}{a}\\ &=\frac {(4 A b+a B) \sqrt {x} (a+b x)^{3/2}}{2 a}-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}}+\frac {1}{4} (3 (4 A b+a B)) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx\\ &=\frac {3}{4} (4 A b+a B) \sqrt {x} \sqrt {a+b x}+\frac {(4 A b+a B) \sqrt {x} (a+b x)^{3/2}}{2 a}-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}}+\frac {1}{8} (3 a (4 A b+a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {3}{4} (4 A b+a B) \sqrt {x} \sqrt {a+b x}+\frac {(4 A b+a B) \sqrt {x} (a+b x)^{3/2}}{2 a}-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}}+\frac {1}{4} (3 a (4 A b+a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {3}{4} (4 A b+a B) \sqrt {x} \sqrt {a+b x}+\frac {(4 A b+a B) \sqrt {x} (a+b x)^{3/2}}{2 a}-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}}+\frac {1}{4} (3 a (4 A b+a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {3}{4} (4 A b+a B) \sqrt {x} \sqrt {a+b x}+\frac {(4 A b+a B) \sqrt {x} (a+b x)^{3/2}}{2 a}-\frac {2 A (a+b x)^{5/2}}{a \sqrt {x}}+\frac {3 a (4 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 82, normalized size = 0.71 \begin {gather*} \frac {\sqrt {a+b x} \left (-8 a A+4 A b x+5 a B x+2 b B x^2\right )}{4 \sqrt {x}}-\frac {3 a (4 A b+a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 158, normalized size = 1.37
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-2 b B \,x^{2}-4 A b x -5 B a x +8 A a \right )}{4 \sqrt {x}}+\frac {\left (\frac {3 a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {b}\, A}{2}+\frac {3 a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) B}{8 \sqrt {b}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\) | \(121\) |
default | \(\frac {\sqrt {b x +a}\, \left (4 B \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, x^{2}+12 A b \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a x +8 A \,b^{\frac {3}{2}} x \sqrt {\left (b x +a \right ) x}+3 B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} x +10 B a x \sqrt {b}\, \sqrt {\left (b x +a \right ) x}-16 A a \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\right )}{8 \sqrt {x}\, \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 129, normalized size = 1.12 \begin {gather*} \frac {3 \, B a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, \sqrt {b}} + \frac {3}{2} \, A a \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \frac {3}{4} \, \sqrt {b x^{2} + a x} B a + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{2 \, x} - \frac {3 \, \sqrt {b x^{2} + a x} A a}{x} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.71, size = 179, normalized size = 1.56 \begin {gather*} \left [\frac {3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, B b^{2} x^{2} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b x}, -\frac {3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, B b^{2} x^{2} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{4 \, b x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 17.45, size = 172, normalized size = 1.50 \begin {gather*} A \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 ) + B \left (\frac {5 a^{\frac {3}{2}} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{4} + \frac {\sqrt {a} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x}{a}}}{2} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 \sqrt {b}}\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^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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