Optimal. Leaf size=292 \[ \frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^4}-\frac {5 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}+\frac {5 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^2}-\frac {10 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}-\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \]
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Rubi [A]
time = 0.10, 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 = {660, 45}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
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 = 0.08, 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 )+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 )+60 b^4 (b d-a e) (d+e x)^4 \log (d+e x)\right )}{12 e^6 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(457\) vs.
\(2(222)=444\).
time = 0.64, size = 458, normalized size = 1.57
method | result | size |
risch | \(\frac {b^{5} x \sqrt {\left (b x +a \right )^{2}}}{e^{5} \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-10 e^{4} a^{2} b^{3}+20 a \,b^{4} d \,e^{3}-10 b^{5} d^{2} e^{2}\right ) x^{3}-5 b^{2} e \left (e^{3} a^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x^{2}-\frac {5 b \left (e^{4} a^{4}+2 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-22 a \,b^{3} d^{3} e +13 b^{4} d^{4}\right ) x}{3}-\frac {3 a^{5} e^{5}+5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}+30 a^{2} b^{3} d^{3} e^{2}-125 a \,b^{4} d^{4} e +77 b^{5} d^{5}}{12 e}\right )}{\left (b x +a \right ) e^{5} \left (e x +d \right )^{4}}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, b^{4} \left (a e -b d \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{6}}\) | \(298\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (240 \ln \left (e x +d \right ) a \,b^{4} d^{3} e^{2} x +240 a \,b^{4} d \,e^{4} x^{3}+12 b^{5} e^{5} x^{5}+440 a \,b^{4} d^{3} e^{2} x +240 \ln \left (e x +d \right ) a \,b^{4} d \,e^{4} x^{3}-360 \ln \left (e x +d \right ) b^{5} d^{3} e^{2} x^{2}-77 b^{5} d^{5}+60 \ln \left (e x +d \right ) a \,b^{4} d^{4} e -60 \ln \left (e x +d \right ) b^{5} d^{5}-180 a^{2} b^{3} d \,e^{4} x^{2}+540 a \,b^{4} d^{2} e^{3} x^{2}-40 a^{3} b^{2} d \,e^{4} x -120 a^{2} b^{3} d^{2} e^{3} x -3 a^{5} e^{5}-240 \ln \left (e x +d \right ) b^{5} d^{4} e x +360 \ln \left (e x +d \right ) a \,b^{4} d^{2} e^{3} x^{2}+60 \ln \left (e x +d \right ) a \,b^{4} e^{5} x^{4}-60 \ln \left (e x +d \right ) b^{5} d \,e^{4} x^{4}+48 b^{5} d \,e^{4} x^{4}-120 a^{2} b^{3} e^{5} x^{3}-48 b^{5} d^{2} e^{3} x^{3}-60 a^{3} b^{2} e^{5} x^{2}-252 b^{5} d^{3} e^{2} x^{2}-20 a^{4} b \,e^{5} x -248 b^{5} d^{4} e x -240 \ln \left (e x +d \right ) b^{5} d^{2} e^{3} x^{3}-5 a^{4} b d \,e^{4}-10 a^{3} b^{2} d^{2} e^{3}-30 a^{2} b^{3} d^{3} e^{2}+125 a \,b^{4} d^{4} e \right )}{12 \left (b x +a \right )^{5} e^{6} \left (e x +d \right )^{4}}\) | \(458\) |
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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Fricas [A]
time = 3.09, size = 388, normalized size = 1.33 \begin {gather*} -\frac {77 \, b^{5} d^{5} - {\left (12 \, b^{5} x^{5} - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}\right )} e^{5} - {\left (48 \, b^{5} d x^{4} + 240 \, a b^{4} d x^{3} - 180 \, a^{2} b^{3} d x^{2} - 40 \, a^{3} b^{2} d x - 5 \, a^{4} b d\right )} e^{4} + 2 \, {\left (24 \, b^{5} d^{2} x^{3} - 270 \, a b^{4} d^{2} x^{2} + 60 \, a^{2} b^{3} d^{2} x + 5 \, a^{3} b^{2} d^{2}\right )} e^{3} + 2 \, {\left (126 \, b^{5} d^{3} x^{2} - 220 \, a b^{4} d^{3} x + 15 \, a^{2} b^{3} d^{3}\right )} e^{2} + {\left (248 \, b^{5} d^{4} x - 125 \, a b^{4} d^{4}\right )} e + 60 \, {\left (b^{5} d^{5} - a b^{4} x^{4} e^{5} + {\left (b^{5} d x^{4} - 4 \, a b^{4} d x^{3}\right )} e^{4} + 2 \, {\left (2 \, b^{5} d^{2} x^{3} - 3 \, a b^{4} d^{2} x^{2}\right )} e^{3} + 2 \, {\left (3 \, b^{5} d^{3} x^{2} - 2 \, a b^{4} d^{3} x\right )} e^{2} + {\left (4 \, b^{5} d^{4} x - a b^{4} d^{4}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{10} + 4 \, d x^{3} e^{9} + 6 \, d^{2} x^{2} e^{8} + 4 \, d^{3} x e^{7} + d^{4} e^{6}\right )}} \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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Giac [A]
time = 1.64, 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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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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