Optimal. Leaf size=249 \[ \frac {2 a^2 (a+b x) (3 A b-5 a B) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a x (a+b x) (A b-2 a B)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x) (A b-3 a B)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4 (A b-a B)}{2 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 (4 A b-5 a B)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.18, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 77} \[ -\frac {a^4 (A b-a B)}{2 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 (4 A b-5 a B)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a x (a+b x) (A b-2 a B)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x) (A b-3 a B)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 (a+b x) (3 A b-5 a B) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {x^4 (A+B x)}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {3 a (-A b+2 a B)}{b^8}+\frac {(A b-3 a B) x}{b^7}+\frac {B x^2}{b^6}-\frac {a^4 (-A b+a B)}{b^8 (a+b x)^3}+\frac {a^3 (-4 A b+5 a B)}{b^8 (a+b x)^2}-\frac {2 a^2 (-3 A b+5 a B)}{b^8 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {a^3 (4 A b-5 a B)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4 (A b-a B)}{2 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a (A b-2 a B) x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-3 a B) x^2 (a+b x)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 (3 A b-5 a B) (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 140, normalized size = 0.56 \[ \frac {-27 a^5 B+3 a^4 b (7 A+2 B x)+3 a^3 b^2 x (2 A+21 B x)+a^2 b^3 x^2 (20 B x-33 A)-12 a^2 (a+b x)^2 (5 a B-3 A b) \log (a+b x)-a b^4 x^3 (12 A+5 B x)+b^5 x^4 (3 A+2 B x)}{6 b^6 (a+b x) \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 197, normalized size = 0.79 \[ \frac {2 \, B b^{5} x^{5} - 27 \, B a^{5} + 21 \, A a^{4} b - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \, {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 3 \, {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 6 \, {\left (B a^{4} b + A a^{3} b^{2}\right )} x - 12 \, {\left (5 \, B a^{5} - 3 \, A a^{4} b + {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
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 = 0.07, size = 217, normalized size = 0.87 \[ \frac {\left (2 B \,b^{5} x^{5}+3 A \,b^{5} x^{4}-5 B a \,b^{4} x^{4}+36 A \,a^{2} b^{3} x^{2} \ln \left (b x +a \right )-12 A a \,b^{4} x^{3}-60 B \,a^{3} b^{2} x^{2} \ln \left (b x +a \right )+20 B \,a^{2} b^{3} x^{3}+72 A \,a^{3} b^{2} x \ln \left (b x +a \right )-33 A \,a^{2} b^{3} x^{2}-120 B \,a^{4} b x \ln \left (b x +a \right )+63 B \,a^{3} b^{2} x^{2}+36 A \,a^{4} b \ln \left (b x +a \right )+6 A \,a^{3} b^{2} x -60 B \,a^{5} \ln \left (b x +a \right )+6 B \,a^{4} b x +21 A \,a^{4} b -27 B \,a^{5}\right ) \left (b x +a \right )}{6 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 303, normalized size = 1.22 \[ \frac {B x^{4}}{3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {7 \, B a x^{3}}{6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {A x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {9 \, B a^{2} x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {5 \, A a x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} - \frac {10 \, B a^{3} \log \left (x + \frac {a}{b}\right )}{b^{6}} + \frac {6 \, A a^{2} \log \left (x + \frac {a}{b}\right )}{b^{5}} + \frac {9 \, B a^{4}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{6}} - \frac {5 \, A a^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} - \frac {20 \, B a^{4} x}{b^{7} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {12 \, A a^{3} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {39 \, B a^{5}}{2 \, b^{8} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, A a^{4}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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