Optimal. Leaf size=200 \[ \frac {35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}}-\frac {35 a \sqrt {x} \sqrt {a+b x} (2 A b-3 a B)}{8 b^5}+\frac {35 x^{3/2} \sqrt {a+b x} (2 A b-3 a B)}{12 b^4}-\frac {7 x^{5/2} \sqrt {a+b x} (2 A b-3 a B)}{3 a b^3}+\frac {2 x^{7/2} (2 A b-3 a B)}{a b^2 \sqrt {a+b x}}+\frac {2 x^{9/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}}+\frac {2 x^{7/2} (2 A b-3 a B)}{a b^2 \sqrt {a+b x}}-\frac {7 x^{5/2} \sqrt {a+b x} (2 A b-3 a B)}{3 a b^3}+\frac {35 x^{3/2} \sqrt {a+b x} (2 A b-3 a B)}{12 b^4}-\frac {35 a \sqrt {x} \sqrt {a+b x} (2 A b-3 a B)}{8 b^5}+\frac {2 x^{9/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}-\frac {\left (2 \left (3 A b-\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{(a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {(7 (2 A b-3 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{a b^2}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {(35 (2 A b-3 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^3}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}-\frac {(35 a (2 A b-3 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b^4}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 80, normalized size = 0.40 \[ \frac {2 x^{9/2} \left ((a+b x) \sqrt {\frac {b x}{a}+1} (3 a B-2 A b) \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {b x}{a}\right )+3 a (A b-a B)\right )}{9 a^2 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 424, normalized size = 2.12 \[ \left [-\frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}, \frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 98.12, size = 382, normalized size = 1.91 \[ \frac {1}{24} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{7}} - \frac {25 \, B a b^{20} {\left | b \right |} - 6 \, A b^{21} {\left | b \right |}}{b^{27}}\right )} + \frac {3 \, {\left (55 \, B a^{2} b^{20} {\left | b \right |} - 26 \, A a b^{21} {\left | b \right |}\right )}}{b^{27}}\right )} + \frac {35 \, {\left (3 \, B a^{3} \sqrt {b} {\left | b \right |} - 2 \, A a^{2} b^{\frac {3}{2}} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{7}} + \frac {4 \, {\left (15 \, B a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} {\left | b \right |} + 24 \, B a^{5} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} {\left | b \right |} - 12 \, A a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {3}{2}} {\left | b \right |} + 13 \, B a^{6} b^{\frac {5}{2}} {\left | b \right |} - 18 \, A a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} {\left | b \right |} - 10 \, A a^{5} b^{\frac {7}{2}} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 406, normalized size = 2.03 \[ \frac {\left (16 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {9}{2}} x^{4}+210 A \,a^{2} b^{3} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-315 B \,a^{3} b^{2} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+24 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {9}{2}} x^{3}-36 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {7}{2}} x^{3}+420 A \,a^{3} b^{2} x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-630 B \,a^{4} b x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-84 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {7}{2}} x^{2}+126 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {5}{2}} x^{2}+210 A \,a^{4} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-315 B \,a^{5} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-560 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {5}{2}} x +840 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {3}{2}} x -420 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {3}{2}}+630 \sqrt {\left (b x +a \right ) x}\, B \,a^{4} \sqrt {b}\right ) \sqrt {x}}{48 \sqrt {\left (b x +a \right ) x}\, \left (b x +a \right )^{\frac {3}{2}} b^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 416, normalized size = 2.08 \[ \frac {B x^{6}}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {3 \, B a x^{5}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{5}}{2 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {35 \, B a^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{16 \, b^{3}} - \frac {35 \, A a^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{24 \, b^{2}} + \frac {21 \, B a^{2} x^{4}}{8 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{3}} - \frac {7 \, A a x^{4}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {35 \, B a^{3} x}{4 \, \sqrt {b x^{2} + a x} b^{5}} - \frac {35 \, A a^{2} x}{6 \, \sqrt {b x^{2} + a x} b^{4}} - \frac {105 \, B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {11}{2}}} + \frac {35 \, A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {9}{2}}} + \frac {35 \, \sqrt {b x^{2} + a x} B a^{2}}{8 \, b^{5}} - \frac {35 \, \sqrt {b x^{2} + a x} A a}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{7/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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