Optimal. Leaf size=133 \[ \frac {(2 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}}-\frac {\sqrt {x} \sqrt {a+b x} (2 A b-5 a B)}{a b^3}+\frac {2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt {a+b x}}+\frac {2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 133, 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 = {78, 47, 50, 63, 217, 206} \[ \frac {2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt {a+b x}}-\frac {\sqrt {x} \sqrt {a+b x} (2 A b-5 a B)}{a b^3}+\frac {(2 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}}+\frac {2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
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)^{5/2}} \, dx &=\frac {2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}-\frac {\left (2 \left (A b-\frac {5 a B}{2}\right )\right ) \int \frac {x^{3/2}}{(a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac {2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(2 A b-5 a B) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{a b^2}\\ &=\frac {2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(2 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{a b^3}+\frac {(2 A b-5 a B) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{2 b^3}\\ &=\frac {2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(2 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{a b^3}+\frac {(2 A b-5 a B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(2 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{a b^3}+\frac {(2 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 )}{b^3}\\ &=\frac {2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(2 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{a b^3}+\frac {(2 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.08, size = 80, normalized size = 0.60 \[ \frac {2 x^{5/2} \left ((a+b x) \sqrt {\frac {b x}{a}+1} (5 a B-2 A b) \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {b x}{a}\right )+5 a (A b-a B)\right )}{15 a^2 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 314, normalized size = 2.36 \[ \left [-\frac {3 \, {\left (5 \, B a^{3} - 2 \, A a^{2} b + {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{6 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {3 \, {\left (5 \, B a^{3} - 2 \, A a^{2} b + {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 108.51, size = 309, normalized size = 2.32 \[ \frac {\sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} B {\left | b \right |}}{b^{5}} + \frac {{\left (5 \, B a \sqrt {b} {\left | b \right |} - 2 \, A b^{\frac {3}{2}} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{2 \, b^{5}} + \frac {4 \, {\left (9 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} {\left | b \right |} + 12 \, B a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} {\left | b \right |} - 6 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {3}{2}} {\left | b \right |} + 7 \, B a^{4} b^{\frac {5}{2}} {\left | b \right |} - 6 \, A a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} {\left | b \right |} - 4 \, A a^{3} b^{\frac {7}{2}} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 315, normalized size = 2.37 \[ \frac {\left (6 A \,b^{3} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-15 B a \,b^{2} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+12 A a \,b^{2} x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-30 B \,a^{2} b x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+6 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {5}{2}} x^{2}+6 A \,a^{2} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-15 B \,a^{3} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-16 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {5}{2}} x +40 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {3}{2}} x -12 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {3}{2}}+30 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} \sqrt {b}\right ) \sqrt {x}}{6 \sqrt {\left (b x +a \right ) x}\, \left (b x +a \right )^{\frac {3}{2}} b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.07, size = 334, normalized size = 2.51 \[ \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{3 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} - \frac {\sqrt {b x^{2} + a x} B a^{2}}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}} + \frac {\sqrt {b x^{2} + a x} A a}{3 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {16 \, \sqrt {b x^{2} + a x} B a}{3 \, {\left (b^{4} x + a b^{3}\right )}} - \frac {7 \, \sqrt {b x^{2} + a x} A}{3 \, {\left (b^{3} x + a b^{2}\right )}} - \frac {5 \, B a \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{2 \, b^{\frac {7}{2}}} + \frac {A \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{3/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________