Optimal. Leaf size=134 \[ -\frac {3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{7/2}}+\frac {3 \sqrt {x} \sqrt {a+b x} (4 A b-5 a B)}{4 b^3}-\frac {x^{3/2} \sqrt {a+b x} (4 A b-5 a B)}{2 a b^2}+\frac {2 x^{5/2} (A b-a B)}{a b \sqrt {a+b x}} \]
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Rubi [A] time = 0.05, antiderivative size = 134, 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 = {78, 50, 63, 217, 206} \begin {gather*} -\frac {x^{3/2} \sqrt {a+b x} (4 A b-5 a B)}{2 a b^2}+\frac {3 \sqrt {x} \sqrt {a+b x} (4 A b-5 a B)}{4 b^3}-\frac {3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{7/2}}+\frac {2 x^{5/2} (A b-a B)}{a b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{(a+b x)^{3/2}} \, dx &=\frac {2 (A b-a B) x^{5/2}}{a b \sqrt {a+b x}}-\frac {\left (2 \left (2 A b-\frac {5 a B}{2}\right )\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{a b}\\ &=\frac {2 (A b-a B) x^{5/2}}{a b \sqrt {a+b x}}-\frac {(4 A b-5 a B) x^{3/2} \sqrt {a+b x}}{2 a b^2}+\frac {(3 (4 A b-5 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{4 b^2}\\ &=\frac {2 (A b-a B) x^{5/2}}{a b \sqrt {a+b x}}+\frac {3 (4 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{4 b^3}-\frac {(4 A b-5 a B) x^{3/2} \sqrt {a+b x}}{2 a b^2}-\frac {(3 a (4 A b-5 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b^3}\\ &=\frac {2 (A b-a B) x^{5/2}}{a b \sqrt {a+b x}}+\frac {3 (4 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{4 b^3}-\frac {(4 A b-5 a B) x^{3/2} \sqrt {a+b x}}{2 a b^2}-\frac {(3 a (4 A b-5 a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=\frac {2 (A b-a B) x^{5/2}}{a b \sqrt {a+b x}}+\frac {3 (4 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{4 b^3}-\frac {(4 A b-5 a B) x^{3/2} \sqrt {a+b x}}{2 a b^2}-\frac {(3 a (4 A b-5 a B)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^3}\\ &=\frac {2 (A b-a B) x^{5/2}}{a b \sqrt {a+b x}}+\frac {3 (4 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{4 b^3}-\frac {(4 A b-5 a B) x^{3/2} \sqrt {a+b x}}{2 a b^2}-\frac {3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 106, normalized size = 0.79 \begin {gather*} \frac {3 a^{3/2} \sqrt {\frac {b x}{a}+1} (5 a B-4 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} \sqrt {x} \left (-15 a^2 B+a b (12 A-5 B x)+2 b^2 x (2 A+B x)\right )}{4 b^{7/2} \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 115, normalized size = 0.86 \begin {gather*} \frac {-15 a^2 B \sqrt {x}+12 a A b \sqrt {x}-5 a b B x^{3/2}+4 A b^2 x^{3/2}+2 b^2 B x^{5/2}}{4 b^3 \sqrt {a+b x}}-\frac {3 \left (5 a^2 B-4 a A b\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{4 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.44, size = 255, normalized size = 1.90 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A 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 (2 \, B b^{3} x^{2} - 15 \, B a^{2} b + 12 \, A a b^{2} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{8 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, B b^{3} x^{2} - 15 \, B a^{2} b + 12 \, A a b^{2} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{4 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 107.23, size = 181, normalized size = 1.35 \begin {gather*} \frac {1}{4} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{5}} - \frac {9 \, B a b^{9} {\left | b \right |} - 4 \, A b^{10} {\left | b \right |}}{b^{14}}\right )} - \frac {3 \, {\left (5 \, B a^{2} \sqrt {b} {\left | b \right |} - 4 \, A 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 )}{8 \, b^{5}} - \frac {4 \, {\left (B a^{3} \sqrt {b} {\left | b \right |} - A a^{2} 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^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 244, normalized size = 1.82 \begin {gather*} -\frac {\left (12 A a \,b^{2} x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-15 B \,a^{2} b x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-4 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {5}{2}} x^{2}+12 A \,a^{2} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-15 B \,a^{3} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-8 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {5}{2}} x +10 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {3}{2}} x -24 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {3}{2}}+30 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} \sqrt {b}\right ) \sqrt {x}}{8 \sqrt {\left (b x +a \right ) x}\, \sqrt {b x +a}\, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 246, normalized size = 1.84 \begin {gather*} -\frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} B a^{2}}{b^{4} x + a b^{3}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{2 \, {\left (b^{3} x + a b^{2}\right )}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{b^{3} x + a b^{2}} + \frac {15 \, B a^{2} \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {3 \, A 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 {3 \, \sqrt {b x^{2} + a x} B a}{4 \, 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 {x^{3/2}\,\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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sympy [A] time = 59.06, size = 182, normalized size = 1.36 \begin {gather*} A \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 ) + B \left (- \frac {15 a^{\frac {3}{2}} \sqrt {x}}{4 b^{3} \sqrt {1 + \frac {b x}{a}}} - \frac {5 \sqrt {a} x^{\frac {3}{2}}}{4 b^{2} \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {7}{2}}} + \frac {x^{\frac {5}{2}}}{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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