Optimal. Leaf size=150 \[ b^{3/2} (5 a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )+\frac {b^2 \sqrt {x} \sqrt {a+b x} (5 a B+2 A b)}{a}-\frac {2 (a+b x)^{5/2} (5 a B+2 A b)}{15 a x^{3/2}}-\frac {2 b (a+b x)^{3/2} (5 a B+2 A b)}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 150, 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} \[ \frac {b^2 \sqrt {x} \sqrt {a+b x} (5 a B+2 A b)}{a}+b^{3/2} (5 a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 (a+b x)^{5/2} (5 a B+2 A b)}{15 a x^{3/2}}-\frac {2 b (a+b x)^{3/2} (5 a B+2 A b)}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}} \]
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 {(a+b x)^{5/2} (A+B x)}{x^{7/2}} \, dx &=-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}}+\frac {\left (2 \left (A b+\frac {5 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{5/2}} \, dx}{5 a}\\ &=-\frac {2 (2 A b+5 a B) (a+b x)^{5/2}}{15 a x^{3/2}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}}+\frac {(b (2 A b+5 a B)) \int \frac {(a+b x)^{3/2}}{x^{3/2}} \, dx}{3 a}\\ &=-\frac {2 b (2 A b+5 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 (2 A b+5 a B) (a+b x)^{5/2}}{15 a x^{3/2}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}}+\frac {\left (b^2 (2 A b+5 a B)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx}{a}\\ &=\frac {b^2 (2 A b+5 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 b (2 A b+5 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 (2 A b+5 a B) (a+b x)^{5/2}}{15 a x^{3/2}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}}+\frac {1}{2} \left (b^2 (2 A b+5 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {b^2 (2 A b+5 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 b (2 A b+5 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 (2 A b+5 a B) (a+b x)^{5/2}}{15 a x^{3/2}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}}+\left (b^2 (2 A b+5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {b^2 (2 A b+5 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 b (2 A b+5 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 (2 A b+5 a B) (a+b x)^{5/2}}{15 a x^{3/2}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}}+\left (b^2 (2 A b+5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {b^2 (2 A b+5 a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 b (2 A b+5 a B) (a+b x)^{3/2}}{3 a \sqrt {x}}-\frac {2 (2 A b+5 a B) (a+b x)^{5/2}}{15 a x^{3/2}}-\frac {2 A (a+b x)^{7/2}}{5 a x^{5/2}}+b^{3/2} (2 A b+5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 76, normalized size = 0.51 \[ \frac {2 \sqrt {a+b x} \left (-\frac {a^2 x (5 a B+2 A b) \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x}{a}\right )}{\sqrt {\frac {b x}{a}+1}}-3 A (a+b x)^3\right )}{15 a x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 217, normalized size = 1.45 \[ \left [\frac {15 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} \sqrt {b} x^{3} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (15 \, B b^{2} x^{3} - 6 \, A a^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{30 \, x^{3}}, -\frac {15 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (15 \, B b^{2} x^{3} - 6 \, A a^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.01, size = 206, normalized size = 1.37 \[ \frac {\sqrt {b x +a}\, \left (30 A \,b^{3} x^{3} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+75 B a \,b^{2} x^{3} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+30 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {5}{2}} x^{3}-92 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {5}{2}} x^{2}-140 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {3}{2}} x^{2}-44 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {3}{2}} x -20 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} \sqrt {b}\, x -12 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} \sqrt {b}\right )}{30 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}\, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.91, size = 244, normalized size = 1.63 \[ \frac {5}{2} \, B a b^{\frac {3}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + A b^{\frac {5}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {35 \, \sqrt {b x^{2} + a x} B a b}{6 \, x} - \frac {38 \, \sqrt {b x^{2} + a x} A b^{2}}{15 \, x} - \frac {5 \, \sqrt {b x^{2} + a x} B a^{2}}{6 \, x^{2}} - \frac {7 \, \sqrt {b x^{2} + a x} A a b}{30 \, x^{2}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{6 \, x^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} A a^{2}}{10 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A b}{3 \, x^{3}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{2 \, x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{5 \, x^{5}} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{x^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 92.56, size = 201, normalized size = 1.34 \[ A \left (- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {22 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{15 x} - \frac {46 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{15} - b^{\frac {5}{2}} \log {\left (\frac {a}{b x} \right )} + 2 b^{\frac {5}{2}} \log {\left (\sqrt {\frac {a}{b x} + 1} + 1 \right )}\right ) + B \left (- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {14 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3} - \frac {5 a b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )}}{2} + 5 a b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x} + 1} + 1 \right )} + b^{\frac {5}{2}} x \sqrt {\frac {a}{b x} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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