Optimal. Leaf size=244 \[ \frac {15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 a^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {20 a^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \[ -\frac {a^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 a^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {20 a^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {x^6}{\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^6}{\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 {5 a}{b^{11}}+\frac {x}{b^{10}}+\frac {a^6}{b^{11} (a+b x)^5}-\frac {6 a^5}{b^{11} (a+b x)^4}+\frac {15 a^4}{b^{11} (a+b x)^3}-\frac {20 a^3}{b^{11} (a+b x)^2}+\frac {15 a^2}{b^{11} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {20 a^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 a^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 106, normalized size = 0.43 \[ \frac {57 a^6+168 a^5 b x+132 a^4 b^2 x^2-32 a^3 b^3 x^3-68 a^2 b^4 x^4+60 a^2 (a+b x)^4 \log (a+b x)-12 a b^5 x^5+2 b^6 x^6}{4 b^7 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 162, normalized size = 0.66 \[ \frac {2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6} + 60 \, {\left (a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + 6 \, a^{4} b^{2} x^{2} + 4 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{4 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \]
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 = 0.24, size = 158, normalized size = 0.65 \[ \frac {\left (2 b^{6} x^{6}+60 a^{2} b^{4} x^{4} \ln \left (b x +a \right )-12 a \,b^{5} x^{5}+240 a^{3} b^{3} x^{3} \ln \left (b x +a \right )-68 a^{2} b^{4} x^{4}+360 a^{4} b^{2} x^{2} \ln \left (b x +a \right )-32 a^{3} b^{3} x^{3}+240 a^{5} b x \ln \left (b x +a \right )+132 a^{4} b^{2} x^{2}+60 a^{6} \ln \left (b x +a \right )+168 a^{5} b x +57 a^{6}\right ) \left (b x +a \right )}{4 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.57, size = 126, normalized size = 0.52 \[ \frac {2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6}}{4 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} + \frac {15 \, a^{2} \log \left (b x + a\right )}{b^{7}} \]
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 \[ \int \frac {x^6}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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