Optimal. Leaf size=295 \[ -\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^6 (a+b x) (d+e x)^2}-\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^6 (a+b x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^6 (a+b x)}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{2 e^6 (a+b x)}+\frac {10 b^3 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x)} \]
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Rubi [A] time = 0.19, antiderivative size = 295, 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 {b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^6 (a+b x)}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{2 e^6 (a+b x)}+\frac {10 b^3 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x)}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{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 \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)^3} \, 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)^3} \, 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 {10 b^8 (b d-a e)^2}{e^5}-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^3}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^2}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)}-\frac {5 b^9 (b d-a e) (d+e x)}{e^5}+\frac {b^{10} (d+e x)^2}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {10 b^3 (b d-a e)^2 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}}{2 e^6 (a+b x) (d+e x)^2}-\frac {5 b (b d-a e)^4 \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) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x)}+\frac {b^5 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}-\frac {10 b^2 (b d-a e)^3 \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 = 0.11, size = 248, normalized size = 0.84 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-3 a^5 e^5-15 a^4 b e^4 (d+2 e x)+30 a^3 b^2 d e^3 (3 d+4 e x)+30 a^2 b^3 e^2 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+15 a b^4 e \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )-60 b^2 (d+e x)^2 (b d-a e)^3 \log (d+e x)+b^5 \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 4.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 416, normalized size = 1.41 \begin {gather*} \frac {2 \, b^{5} e^{5} x^{5} - 27 \, b^{5} d^{5} + 105 \, a b^{4} d^{4} e - 150 \, a^{2} b^{3} d^{3} e^{2} + 90 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \, {\left (b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} - 3 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{5} d^{3} e^{2} - 55 \, a b^{4} d^{2} e^{3} + 40 \, a^{2} b^{3} d e^{4}\right )} x^{2} + 6 \, {\left (b^{5} d^{4} e + 5 \, a b^{4} d^{3} e^{2} - 20 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} d^{5} - 3 \, a b^{4} d^{4} e + 3 \, a^{2} b^{3} d^{3} e^{2} - a^{3} b^{2} d^{2} e^{3} + {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{5} d^{4} e - 3 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3} - a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 376, normalized size = 1.27 \begin {gather*} -10 \, {\left (b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, b^{5} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) - 9 \, b^{5} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{5} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) - 90 \, a b^{4} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 60 \, a^{2} b^{3} x e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-9\right )} - \frac {{\left (9 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 35 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 50 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 30 \, 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 ) + a^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 10 \, {\left (b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 4 \, 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 ) - 4 \, 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 )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 502, normalized size = 1.70 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (2 b^{5} e^{5} x^{5}+15 a \,b^{4} e^{5} x^{4}-5 b^{5} d \,e^{4} x^{4}+60 a^{3} b^{2} e^{5} x^{2} \ln \left (e x +d \right )-180 a^{2} b^{3} d \,e^{4} x^{2} \ln \left (e x +d \right )+60 a^{2} b^{3} e^{5} x^{3}+180 a \,b^{4} d^{2} e^{3} x^{2} \ln \left (e x +d \right )-60 a \,b^{4} d \,e^{4} x^{3}-60 b^{5} d^{3} e^{2} x^{2} \ln \left (e x +d \right )+20 b^{5} d^{2} e^{3} x^{3}+120 a^{3} b^{2} d \,e^{4} x \ln \left (e x +d \right )-360 a^{2} b^{3} d^{2} e^{3} x \ln \left (e x +d \right )+120 a^{2} b^{3} d \,e^{4} x^{2}+360 a \,b^{4} d^{3} e^{2} x \ln \left (e x +d \right )-165 a \,b^{4} d^{2} e^{3} x^{2}-120 b^{5} d^{4} e x \ln \left (e x +d \right )+63 b^{5} d^{3} e^{2} x^{2}-30 a^{4} b \,e^{5} x +60 a^{3} b^{2} d^{2} e^{3} \ln \left (e x +d \right )+120 a^{3} b^{2} d \,e^{4} x -180 a^{2} b^{3} d^{3} e^{2} \ln \left (e x +d \right )-120 a^{2} b^{3} d^{2} e^{3} x +180 a \,b^{4} d^{4} e \ln \left (e x +d \right )+30 a \,b^{4} d^{3} e^{2} x -60 b^{5} d^{5} \ln \left (e x +d \right )+6 b^{5} d^{4} e x -3 a^{5} e^{5}-15 a^{4} b d \,e^{4}+90 a^{3} b^{2} d^{2} e^{3}-150 a^{2} b^{3} d^{3} e^{2}+105 a \,b^{4} d^{4} e -27 b^{5} d^{5}\right )}{6 \left (b x +a \right )^{5} \left (e x +d \right )^{2} 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 )}^3} \,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 )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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