Optimal. Leaf size=98 \[ \frac {2 (A b-a B) x^{3/2}}{a b \sqrt {a+b x}}-\frac {(2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{a b^2}+\frac {(2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {(2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}}-\frac {\sqrt {x} \sqrt {a+b x} (2 A b-3 a B)}{a b^2}+\frac {2 x^{3/2} (A b-a B)}{a b \sqrt {a+b 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 {\sqrt {x} (A+B x)}{(a+b x)^{3/2}} \, dx &=\frac {2 (A b-a B) x^{3/2}}{a b \sqrt {a+b x}}-\frac {\left (2 \left (A b-\frac {3 a B}{2}\right )\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{a b}\\ &=\frac {2 (A b-a B) x^{3/2}}{a b \sqrt {a+b x}}-\frac {(2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{a b^2}+\frac {(2 A b-3 a B) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{2 b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{a b \sqrt {a+b x}}-\frac {(2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{a b^2}+\frac {(2 A b-3 a B) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{a b \sqrt {a+b x}}-\frac {(2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{a b^2}+\frac {(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 )}{b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{a b \sqrt {a+b x}}-\frac {(2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{a b^2}+\frac {(2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 70, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x} (-2 A b+3 a B+b B x)}{b^2 \sqrt {a+b x}}+\frac {(-2 A b+3 a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(200\) vs.
\(2(82)=164\).
time = 0.09, size = 201, normalized size = 2.05
method | result | size |
risch | \(\frac {B \sqrt {b x +a}\, \sqrt {x}}{b^{2}}+\frac {\left (\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) A}{b^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) B a}{2 b^{\frac {5}{2}}}-\frac {2 \sqrt {b \left (x +\frac {a}{b}\right )^{2}-a \left (x +\frac {a}{b}\right )}\, A}{b^{2} \left (x +\frac {a}{b}\right )}+\frac {2 \sqrt {b \left (x +\frac {a}{b}\right )^{2}-a \left (x +\frac {a}{b}\right )}\, B a}{b^{3} \left (x +\frac {a}{b}\right )}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\) | \(177\) |
default | \(\frac {\left (2 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) b^{2} x -3 B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a b x +2 B \,b^{\frac {3}{2}} x \sqrt {\left (b x +a \right ) x}+2 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a b -4 A \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}-3 B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2}+6 B a \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\right ) \sqrt {x}}{2 \sqrt {\left (b x +a \right ) x}\, b^{\frac {5}{2}} \sqrt {b x +a}}\) | \(201\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 134, normalized size = 1.37 \begin {gather*} \frac {2 \, \sqrt {b x^{2} + a x} B a}{b^{3} x + a b^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{b^{2} x + a b} - \frac {3 \, B a \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {A \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {\sqrt {b x^{2} + a x} B}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.16, size = 195, normalized size = 1.99 \begin {gather*} \left [-\frac {{\left (3 \, B a^{2} - 2 \, A a b + {\left (3 \, B a b - 2 \, A b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (B b^{2} x + 3 \, B a b - 2 \, A b^{2}\right )} \sqrt {b x + a} \sqrt {x}}{2 \, {\left (b^{4} x + a b^{3}\right )}}, \frac {{\left (3 \, B a^{2} - 2 \, A a b + {\left (3 \, B a b - 2 \, A b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (B b^{2} x + 3 \, B a b - 2 \, A b^{2}\right )} \sqrt {b x + a} \sqrt {x}}{b^{4} x + a b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.31, size = 122, normalized size = 1.24 \begin {gather*} A \left (\frac {2 \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {2 \sqrt {x}}{\sqrt {a} b \sqrt {1 + \frac {b x}{a}}}\right ) + B \left (\frac {3 \sqrt {a} \sqrt {x}}{b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {5}{2}}} + \frac {x^{\frac {3}{2}}}{\sqrt {a} b \sqrt {1 + \frac {b x}{a}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 21.37, size = 144, normalized size = 1.47 \begin {gather*} \frac {\sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} B {\left | b \right |}}{b^{4}} + \frac {{\left (3 \, B a \sqrt {b} {\left | b \right |} - 2 \, A b^{\frac {3}{2}} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{2 \, b^{4}} + \frac {4 \, {\left (B a^{2} \sqrt {b} {\left | b \right |} - A a b^{\frac {3}{2}} {\left | b \right |}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{3}} \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 )}{{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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