Optimal. Leaf size=245 \[ -\frac {a^2 (3 A b-5 a B)}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a (2 A b-5 a B)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-5 a B) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4 (A b-a B)}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 (4 A b-5 a B)}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.18, antiderivative size = 245, 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} \[ -\frac {a^4 (A b-a B)}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 (4 A b-5 a B)}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^2 (3 A b-5 a B)}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a (2 A b-5 a B)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-5 a B) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 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 )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x^4 (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {B}{b^{10}}-\frac {a^4 (-A b+a B)}{b^{10} (a+b x)^5}+\frac {a^3 (-4 A b+5 a B)}{b^{10} (a+b x)^4}-\frac {2 a^2 (-3 A b+5 a B)}{b^{10} (a+b x)^3}+\frac {2 a (-2 A b+5 a B)}{b^{10} (a+b x)^2}+\frac {A b-5 a B}{b^{10} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 a (2 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)}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 (4 A b-5 a B)}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^2 (3 A b-5 a B)}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(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 = 0.07, size = 127, normalized size = 0.52 \[ \frac {-77 a^5 B+a^4 b (25 A-248 B x)+4 a^3 b^2 x (22 A-63 B x)+12 a^2 b^3 x^2 (9 A-4 B x)+48 a b^4 x^3 (A+B x)+12 (a+b x)^4 (A b-5 a B) \log (a+b x)+12 b^5 B x^5}{12 b^6 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 252, normalized size = 1.03 \[ \frac {12 \, B b^{5} x^{5} + 48 \, B a b^{4} x^{4} - 77 \, B a^{5} + 25 \, A a^{4} b - 48 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 36 \, {\left (7 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 8 \, {\left (31 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x - 12 \, {\left (5 \, B a^{5} - A a^{4} b + {\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \, {\left (5 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \, {\left (5 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (5 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \]
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 = 273, normalized size = 1.11 \[ \frac {\left (12 A \,b^{5} x^{4} \ln \left (b x +a \right )-60 B a \,b^{4} x^{4} \ln \left (b x +a \right )+12 B \,b^{5} x^{5}+48 A a \,b^{4} x^{3} \ln \left (b x +a \right )-240 B \,a^{2} b^{3} x^{3} \ln \left (b x +a \right )+48 B a \,b^{4} x^{4}+72 A \,a^{2} b^{3} x^{2} \ln \left (b x +a \right )+48 A a \,b^{4} x^{3}-360 B \,a^{3} b^{2} x^{2} \ln \left (b x +a \right )-48 B \,a^{2} b^{3} x^{3}+48 A \,a^{3} b^{2} x \ln \left (b x +a \right )+108 A \,a^{2} b^{3} x^{2}-240 B \,a^{4} b x \ln \left (b x +a \right )-252 B \,a^{3} b^{2} x^{2}+12 A \,a^{4} b \ln \left (b x +a \right )+88 A \,a^{3} b^{2} x -60 B \,a^{5} \ln \left (b x +a \right )-248 B \,a^{4} b x +25 A \,a^{4} b -77 B \,a^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 211, normalized size = 0.86 \[ \frac {1}{12} \, B {\left (\frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}} - \frac {60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac {1}{12} \, A {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} \]
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 \[ \int \frac {x^4\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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