Optimal. Leaf size=225 \[ -\frac {a^5 (12 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{9/2}}+\frac {a^4 \sqrt {x} \sqrt {a+b x} (12 A b-7 a B)}{512 b^4}-\frac {a^3 x^{3/2} \sqrt {a+b x} (12 A b-7 a B)}{768 b^3}+\frac {a^2 x^{5/2} \sqrt {a+b x} (12 A b-7 a B)}{960 b^2}+\frac {a x^{7/2} \sqrt {a+b x} (12 A b-7 a B)}{160 b}+\frac {x^{7/2} (a+b x)^{3/2} (12 A b-7 a B)}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b} \]
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Rubi [A] time = 0.10, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {80, 50, 63, 217, 206} \begin {gather*} -\frac {a^3 x^{3/2} \sqrt {a+b x} (12 A b-7 a B)}{768 b^3}+\frac {a^2 x^{5/2} \sqrt {a+b x} (12 A b-7 a B)}{960 b^2}+\frac {a^4 \sqrt {x} \sqrt {a+b x} (12 A b-7 a B)}{512 b^4}-\frac {a^5 (12 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{9/2}}+\frac {a x^{7/2} \sqrt {a+b x} (12 A b-7 a B)}{160 b}+\frac {x^{7/2} (a+b x)^{3/2} (12 A b-7 a B)}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx &=\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {\left (6 A b-\frac {7 a B}{2}\right ) \int x^{5/2} (a+b x)^{3/2} \, dx}{6 b}\\ &=\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {(a (12 A b-7 a B)) \int x^{5/2} \sqrt {a+b x} \, dx}{40 b}\\ &=\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {\left (a^2 (12 A b-7 a B)\right ) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{320 b}\\ &=\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^3 (12 A b-7 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{384 b^2}\\ &=-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}+\frac {\left (a^4 (12 A b-7 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{512 b^3}\\ &=\frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^5 (12 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{1024 b^4}\\ &=\frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^5 (12 A b-7 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^4}\\ &=\frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {\left (a^5 (12 A b-7 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^4}\\ &=\frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {a^5 (12 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 151, normalized size = 0.67 \begin {gather*} \frac {\sqrt {a+b x} (12 A b-7 a B) \left (\sqrt {b} \sqrt {x} \sqrt {\frac {b x}{a}+1} \left (15 a^4-10 a^3 b x+8 a^2 b^2 x^2+176 a b^3 x^3+128 b^4 x^4\right )-15 a^{9/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{7680 b^{9/2} \sqrt {\frac {b x}{a}+1}}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 201, normalized size = 0.89 \begin {gather*} \frac {\left (12 a^5 A b-7 a^6 B\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{512 b^{9/2}}+\frac {\sqrt {a+b x} \left (-105 a^5 B \sqrt {x}+180 a^4 A b \sqrt {x}+70 a^4 b B x^{3/2}-120 a^3 A b^2 x^{3/2}-56 a^3 b^2 B x^{5/2}+96 a^2 A b^3 x^{5/2}+48 a^2 b^3 B x^{7/2}+2112 a A b^4 x^{7/2}+1664 a b^4 B x^{9/2}+1536 A b^5 x^{9/2}+1280 b^5 B x^{11/2}\right )}{7680 b^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.89, size = 343, normalized size = 1.52 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15360 \, b^{5}}, -\frac {15 \, {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{7680 \, b^{5}}\right ] \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 302, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (-2560 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {11}{2}} x^{5}-3072 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {11}{2}} x^{4}-3328 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {9}{2}} x^{4}-4224 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {9}{2}} x^{3}-96 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {7}{2}} x^{3}-192 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {7}{2}} x^{2}+112 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {5}{2}} x^{2}+180 A \,a^{5} 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^{6} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+240 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {5}{2}} x -140 \sqrt {\left (b x +a \right ) x}\, B \,a^{4} b^{\frac {3}{2}} x -360 \sqrt {\left (b x +a \right ) x}\, A \,a^{4} b^{\frac {3}{2}}+210 \sqrt {\left (b x +a \right ) x}\, B \,a^{5} \sqrt {b}\right ) \sqrt {x}}{15360 \sqrt {\left (b x +a \right ) x}\, b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 280, normalized size = 1.24 \begin {gather*} -\frac {7 \, \sqrt {b x^{2} + a x} B a^{4} x}{256 \, b^{3}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2} x}{96 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} A a^{3} x}{64 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B x}{6 \, b} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a x}{8 \, b} + \frac {7 \, B a^{6} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {9}{2}}} - \frac {3 \, A a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{5}}{512 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{3}}{192 \, b^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} A a^{4}}{128 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} B a}{60 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a^{2}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{5 \, b} \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 = -0.00 \begin {gather*} \int x^{5/2}\,\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 [C] time = 134.42, size = 2825, normalized size = 12.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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