Optimal. Leaf size=292 \[ \frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}-\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)} \]
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Rubi [A] time = 0.15, antiderivative size = 292, 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 {10 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)}+\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (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)^5} \, 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)^5} \, 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^{10}}{e^5}-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^5}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^4}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^3}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^2}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^4}-\frac {5 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}+\frac {5 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^2}-\frac {10 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)}-\frac {5 b^4 (b d-a e) \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 = 243, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a^5 e^5+5 a^4 b e^4 (d+4 e x)+10 a^3 b^2 e^3 \left (d^2+4 d e x+6 e^2 x^2\right )+30 a^2 b^3 e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a b^4 d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+60 b^4 (d+e x)^4 (b d-a e) \log (d+e x)+b^5 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )}{12 e^6 (a+b x) (d+e x)^4} \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 = 0.39, size = 412, normalized size = 1.41 \begin {gather*} \frac {12 \, b^{5} e^{5} x^{5} + 48 \, b^{5} d e^{4} x^{4} - 77 \, b^{5} d^{5} + 125 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 24 \, {\left (2 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 5 \, a^{2} b^{3} e^{5}\right )} x^{3} - 12 \, {\left (21 \, b^{5} d^{3} e^{2} - 45 \, a b^{4} d^{2} e^{3} + 15 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} - 4 \, {\left (62 \, b^{5} d^{4} e - 110 \, a b^{4} d^{3} e^{2} + 30 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} d^{5} - a b^{4} d^{4} e + {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \, {\left (b^{5} d^{2} e^{3} - a b^{4} d e^{4}\right )} x^{3} + 6 \, {\left (b^{5} d^{3} e^{2} - a b^{4} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{5} d^{4} e - a b^{4} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 370, normalized size = 1.27 \begin {gather*} b^{5} x e^{\left (-5\right )} \mathrm {sgn}\left (b x + a\right ) - 5 \, {\left (b^{5} d \mathrm {sgn}\left (b x + a\right ) - a b^{4} e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (77 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 125 \, 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 ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 120 \, {\left (b^{5} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{4} d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{3} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 60 \, {\left (5 \, b^{5} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 9 \, a b^{4} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{3} d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{2} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 20 \, {\left (13 \, b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 22 \, a b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{4} b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-6\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 458, normalized size = 1.57 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 a \,b^{4} e^{5} x^{4} \ln \left (e x +d \right )-60 b^{5} d \,e^{4} x^{4} \ln \left (e x +d \right )+12 b^{5} e^{5} x^{5}+240 a \,b^{4} d \,e^{4} x^{3} \ln \left (e x +d \right )-240 b^{5} d^{2} e^{3} x^{3} \ln \left (e x +d \right )+48 b^{5} d \,e^{4} x^{4}-120 a^{2} b^{3} e^{5} x^{3}+360 a \,b^{4} d^{2} e^{3} x^{2} \ln \left (e x +d \right )+240 a \,b^{4} d \,e^{4} x^{3}-360 b^{5} d^{3} e^{2} x^{2} \ln \left (e x +d \right )-48 b^{5} d^{2} e^{3} x^{3}-60 a^{3} b^{2} e^{5} x^{2}-180 a^{2} b^{3} d \,e^{4} x^{2}+240 a \,b^{4} d^{3} e^{2} x \ln \left (e x +d \right )+540 a \,b^{4} d^{2} e^{3} x^{2}-240 b^{5} d^{4} e x \ln \left (e x +d \right )-252 b^{5} d^{3} e^{2} x^{2}-20 a^{4} b \,e^{5} x -40 a^{3} b^{2} d \,e^{4} x -120 a^{2} b^{3} d^{2} e^{3} x +60 a \,b^{4} d^{4} e \ln \left (e x +d \right )+440 a \,b^{4} d^{3} e^{2} x -60 b^{5} d^{5} \ln \left (e x +d \right )-248 b^{5} d^{4} e x -3 a^{5} e^{5}-5 a^{4} b d \,e^{4}-10 a^{3} b^{2} d^{2} e^{3}-30 a^{2} b^{3} d^{3} e^{2}+125 a \,b^{4} d^{4} e -77 b^{5} d^{5}\right )}{12 \left (b x +a \right )^{5} \left (e x +d \right )^{4} 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 )}^5} \,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 )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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