Optimal. Leaf size=89 \[ -\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+2 b^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 b B \sqrt {a+b x}}{\sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 47, 63, 217, 206} \begin {gather*} -\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+2 b^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 b B \sqrt {a+b x}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^{7/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+B \int \frac {(a+b x)^{3/2}}{x^{5/2}} \, dx\\ &=-\frac {2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+(b B) \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx\\ &=-\frac {2 b B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+\left (b^2 B\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=-\frac {2 b B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+\left (2 b^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+\left (2 b^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=-\frac {2 b B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac {2 A (a+b x)^{5/2}}{5 a x^{5/2}}+2 b^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.08, size = 77, normalized size = 0.87 \begin {gather*} \frac {2 \sqrt {a+b x} \left (-\frac {a^3 B \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};-\frac {b x}{a}\right )}{\sqrt {\frac {b x}{a}+1}}-(a+b x)^2 (A b-a B)\right )}{5 a b x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 90, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (3 a^2 A+5 a^2 B x+6 a A b x+20 a b B x^2+3 A b^2 x^2\right )}{15 a x^{5/2}}-2 b^{3/2} B \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 181, normalized size = 2.03 \begin {gather*} \left [\frac {15 \, B a b^{\frac {3}{2}} x^{3} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (3 \, A a^{2} + {\left (20 \, B a b + 3 \, A b^{2}\right )} x^{2} + {\left (5 \, B a^{2} + 6 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15 \, a x^{3}}, -\frac {2 \, {\left (15 \, B a \sqrt {-b} b x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (3 \, A a^{2} + {\left (20 \, B a b + 3 \, A b^{2}\right )} x^{2} + {\left (5 \, B a^{2} + 6 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}\right )}}{15 \, a x^{3}}\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 [B] time = 0.02, size = 156, normalized size = 1.75 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (-15 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 )+6 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {5}{2}} x^{2}+40 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {3}{2}} x^{2}+12 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {3}{2}} x +10 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} \sqrt {b}\, x +6 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} \sqrt {b}\right )}{15 \sqrt {\left (b x +a \right ) x}\, a \sqrt {b}\, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.94, size = 158, normalized size = 1.78 \begin {gather*} B b^{\frac {3}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {7 \, \sqrt {b x^{2} + a x} B b}{3 \, x} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{5 \, a x} - \frac {\sqrt {b x^{2} + a x} B a}{3 \, x^{2}} + \frac {\sqrt {b x^{2} + a x} A b}{5 \, x^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{3 \, x^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{5 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{x^{4}} \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 = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{x^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 70.27, size = 141, normalized size = 1.58 \begin {gather*} A \left (- \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{5 x} - \frac {2 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a}\right ) + B \left (- \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {8 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3} - b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )} + 2 b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x} + 1} + 1 \right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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