Optimal. Leaf size=300 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2} \]
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Rubi [A] time = 0.14, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \begin {gather*} \frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2}+\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^6} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^6}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^5}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^4}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^3}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^2}+\frac {b^{10}}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}-\frac {5 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^4}+\frac {10 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}-\frac {5 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^2}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 196, normalized size = 0.65 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left ((b d-a e) \left (12 a^4 e^4+3 a^3 b e^3 (9 d+25 e x)+a^2 b^2 e^2 \left (47 d^2+175 d e x+200 e^2 x^2\right )+a b^3 e \left (77 d^3+325 d^2 e x+500 d e^2 x^2+300 e^3 x^3\right )+b^4 \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 b^5 (d+e x)^5 \log (d+e x)\right )}{60 e^6 (a+b x) (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 373, normalized size = 1.24 \begin {gather*} \frac {137 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} + 300 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \, {\left (3 \, b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 100 \, {\left (11 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} - 3 \, a^{2} b^{3} d e^{4} - 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \, {\left (25 \, b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 3 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, b^{5} d e^{4} x^{4} + 10 \, b^{5} d^{2} e^{3} x^{3} + 10 \, b^{5} d^{3} e^{2} x^{2} + 5 \, b^{5} d^{4} e x + b^{5} d^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 378, normalized size = 1.26 \begin {gather*} b^{5} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (300 \, {\left (b^{5} d e^{3} \mathrm {sgn}\left (b x + a\right ) - a b^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 300 \, {\left (3 \, b^{5} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{4} d e^{3} \mathrm {sgn}\left (b x + a\right ) - a^{2} b^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 100 \, {\left (11 \, b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{2} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{3} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 25 \, {\left (25 \, b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x + {\left (137 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 60 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 383, normalized size = 1.28 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 b^{5} e^{5} x^{5} \ln \left (e x +d \right )+300 b^{5} d \,e^{4} x^{4} \ln \left (e x +d \right )-300 a \,b^{4} e^{5} x^{4}+600 b^{5} d^{2} e^{3} x^{3} \ln \left (e x +d \right )+300 b^{5} d \,e^{4} x^{4}-300 a^{2} b^{3} e^{5} x^{3}-600 a \,b^{4} d \,e^{4} x^{3}+600 b^{5} d^{3} e^{2} x^{2} \ln \left (e x +d \right )+900 b^{5} d^{2} e^{3} x^{3}-200 a^{3} b^{2} e^{5} x^{2}-300 a^{2} b^{3} d \,e^{4} x^{2}-600 a \,b^{4} d^{2} e^{3} x^{2}+300 b^{5} d^{4} e x \ln \left (e x +d \right )+1100 b^{5} d^{3} e^{2} x^{2}-75 a^{4} b \,e^{5} x -100 a^{3} b^{2} d \,e^{4} x -150 a^{2} b^{3} d^{2} e^{3} x -300 a \,b^{4} d^{3} e^{2} x +60 b^{5} d^{5} \ln \left (e x +d \right )+625 b^{5} d^{4} e x -12 a^{5} e^{5}-15 a^{4} b d \,e^{4}-20 a^{3} b^{2} d^{2} e^{3}-30 a^{2} b^{3} d^{3} e^{2}-60 a \,b^{4} d^{4} e +137 b^{5} d^{5}\right )}{60 \left (b x +a \right )^{5} \left (e x +d \right )^{5} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 = -0.00 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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