Optimal. Leaf size=169 \[ -\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{9/2}}+\frac {5 \sqrt {x} \sqrt {a+b x} (4 A b-7 a B)}{4 b^4}-\frac {5 x^{3/2} \sqrt {a+b x} (4 A b-7 a B)}{6 a b^3}+\frac {2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt {a+b x}}+\frac {2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt {a+b x}}-\frac {5 x^{3/2} \sqrt {a+b x} (4 A b-7 a B)}{6 a b^3}+\frac {5 \sqrt {x} \sqrt {a+b x} (4 A b-7 a B)}{4 b^4}-\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{9/2}}+\frac {2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx &=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}-\frac {\left (2 \left (2 A b-\frac {7 a B}{2}\right )\right ) \int \frac {x^{5/2}}{(a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(5 (4 A b-7 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{3 a b^2}\\ &=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}+\frac {(5 (4 A b-7 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{4 b^3}\\ &=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {(5 a (4 A b-7 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b^4}\\ &=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {(5 a (4 A b-7 a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^4}\\ &=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {(5 a (4 A b-7 a B)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^4}\\ &=\frac {2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt {a+b x}}+\frac {5 (4 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{4 b^4}-\frac {5 (4 A b-7 a B) x^{3/2} \sqrt {a+b x}}{6 a b^3}-\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 80, normalized size = 0.47 \[ \frac {2 x^{7/2} \left ((a+b x) \sqrt {\frac {b x}{a}+1} (7 a B-4 A b) \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};-\frac {b x}{a}\right )+7 a (A b-a B)\right )}{21 a^2 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 373, normalized size = 2.21 \[ \left [-\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (6 \, B b^{4} x^{3} - 105 \, B a^{3} b + 60 \, A a^{2} b^{2} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{2} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (6 \, B b^{4} x^{3} - 105 \, B a^{3} b + 60 \, A a^{2} b^{2} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{2} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{12 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 107.31, size = 346, normalized size = 2.05 \[ \frac {1}{4} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{6}} - \frac {13 \, B a b^{11} {\left | b \right |} - 4 \, A b^{12} {\left | b \right |}}{b^{17}}\right )} - \frac {5 \, {\left (7 \, B a^{2} \sqrt {b} {\left | b \right |} - 4 \, A a b^{\frac {3}{2}} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{8 \, b^{6}} - \frac {4 \, {\left (12 \, B a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} {\left | b \right |} + 18 \, B a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} {\left | b \right |} - 9 \, A a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {3}{2}} {\left | b \right |} + 10 \, B a^{5} b^{\frac {5}{2}} {\left | b \right |} - 12 \, A a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} {\left | b \right |} - 7 \, A a^{4} b^{\frac {7}{2}} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 362, normalized size = 2.14 \[ -\frac {\left (60 A a \,b^{3} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-105 B \,a^{2} b^{2} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-12 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {7}{2}} x^{3}+120 A \,a^{2} b^{2} x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-210 B \,a^{3} b x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-24 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {7}{2}} x^{2}+42 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {5}{2}} x^{2}+60 A \,a^{3} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-105 B \,a^{4} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-160 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {5}{2}} x +280 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {3}{2}} x -120 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {3}{2}}+210 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} \sqrt {b}\right ) \sqrt {x}}{24 \sqrt {\left (b x +a \right ) x}\, \left (b x +a \right )^{\frac {3}{2}} b^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.09, size = 517, normalized size = 3.06 \[ -\frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B a}{b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2}}{6 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} + \frac {5 \, \sqrt {b x^{2} + a x} B a^{3}}{6 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{2 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{6 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {115 \, \sqrt {b x^{2} + a x} B a^{2}}{12 \, {\left (b^{5} x + a b^{4}\right )}} + \frac {35 \, \sqrt {b x^{2} + a x} A a}{6 \, {\left (b^{4} x + a b^{3}\right )}} + \frac {35 \, B a^{2} \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{8 \, b^{\frac {9}{2}}} - \frac {5 \, A a \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{2 \, b^{\frac {7}{2}}} \]
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 \[ \int \frac {x^{5/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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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