Optimal. Leaf size=188 \[ \frac {x^4 (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 a B}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a^2 B}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 B}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {770, 78, 43} \[ \frac {x^4 (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 a B}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a^2 B}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 B}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 78
Rule 770
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x^3 (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^4}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 B \left (a b+b^2 x\right )\right ) \int \frac {x^3}{\left (a b+b^2 x\right )^4} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^4}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 B \left (a b+b^2 x\right )\right ) \int \left (-\frac {a^3}{b^7 (a+b x)^4}+\frac {3 a^2}{b^7 (a+b x)^3}-\frac {3 a}{b^7 (a+b x)^2}+\frac {1}{b^7 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 a B}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^4}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 B}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a^2 B}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 103, normalized size = 0.55 \[ \frac {25 a^4 B+a^3 (88 b B x-3 A b)-12 a^2 b^2 x (A-9 B x)+6 a b^3 x^2 (8 B x-3 A)+12 B (a+b x)^4 \log (a+b x)-12 A b^4 x^3}{12 b^5 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 175, normalized size = 0.93 \[ \frac {25 \, B a^{4} - 3 \, A a^{3} b + 12 \, {\left (4 \, B a b^{3} - A b^{4}\right )} x^{3} + 18 \, {\left (6 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 4 \, {\left (22 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x + 12 \, {\left (B b^{4} x^{4} + 4 \, B a b^{3} x^{3} + 6 \, B a^{2} b^{2} x^{2} + 4 \, B a^{3} b x + B a^{4}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 168, normalized size = 0.89 \[ -\frac {\left (-12 B \,b^{4} x^{4} \ln \left (b x +a \right )-48 B a \,b^{3} x^{3} \ln \left (b x +a \right )+12 A \,b^{4} x^{3}-72 B \,a^{2} b^{2} x^{2} \ln \left (b x +a \right )-48 B a \,b^{3} x^{3}+18 A a \,b^{3} x^{2}-48 B \,a^{3} b x \ln \left (b x +a \right )-108 B \,a^{2} b^{2} x^{2}+12 A \,a^{2} b^{2} x -12 B \,a^{4} \ln \left (b x +a \right )-88 B \,a^{3} b x +3 A \,a^{3} b -25 B \,a^{4}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 201, normalized size = 1.07 \[ \frac {1}{12} \, B {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} - \frac {1}{12} \, A {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} \]
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 \[ \int \frac {x^3\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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