Optimal. Leaf size=356 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{6 e^7 (a+b x) (d+e x)^6}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}+\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^2}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3} \]
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Rubi [A] time = 0.21, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^2}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{6 e^7 (a+b x) (d+e x)^6}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^7} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^7}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac {b^6}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 258, normalized size = 0.72 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left ((b d-a e) \left (10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+a^2 b^3 e^2 \left (57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right )+a b^4 e \left (87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )+60 b^6 (d+e x)^6 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.26, 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 = 0.42, size = 492, normalized size = 1.38 \begin {gather*} \frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 507, normalized size = 1.42 \begin {gather*} b^{6} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (360 \, {\left (b^{6} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a b^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{2} b^{4} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{2} b^{4} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{3} b^{3} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 12 \, a b^{5} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{2} b^{4} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{4} b^{2} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 60 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x + {\left (147 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 60 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 507, normalized size = 1.42 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 b^{6} e^{6} x^{6} \ln \left (e x +d \right )+360 b^{6} d \,e^{5} x^{5} \ln \left (e x +d \right )-360 a \,b^{5} e^{6} x^{5}+900 b^{6} d^{2} e^{4} x^{4} \ln \left (e x +d \right )+360 b^{6} d \,e^{5} x^{5}-450 a^{2} b^{4} e^{6} x^{4}-900 a \,b^{5} d \,e^{5} x^{4}+1200 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )+1350 b^{6} d^{2} e^{4} x^{4}-400 a^{3} b^{3} e^{6} x^{3}-600 a^{2} b^{4} d \,e^{5} x^{3}-1200 a \,b^{5} d^{2} e^{4} x^{3}+900 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )+2200 b^{6} d^{3} e^{3} x^{3}-225 a^{4} b^{2} e^{6} x^{2}-300 a^{3} b^{3} d \,e^{5} x^{2}-450 a^{2} b^{4} d^{2} e^{4} x^{2}-900 a \,b^{5} d^{3} e^{3} x^{2}+360 b^{6} d^{5} e x \ln \left (e x +d \right )+1875 b^{6} d^{4} e^{2} x^{2}-72 a^{5} b \,e^{6} x -90 a^{4} b^{2} d \,e^{5} x -120 a^{3} b^{3} d^{2} e^{4} x -180 a^{2} b^{4} d^{3} e^{3} x -360 a \,b^{5} d^{4} e^{2} x +60 b^{6} d^{6} \ln \left (e x +d \right )+822 b^{6} d^{5} e x -10 a^{6} e^{6}-12 a^{5} b d \,e^{5}-15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}-60 a \,b^{5} d^{5} e +147 b^{6} d^{6}\right )}{60 \left (b x +a \right )^{5} \left (e x +d \right )^{6} e^{7}} \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+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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