Optimal. Leaf size=146 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^6}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^3 (a+b x) (d+e x)^7} \]
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Rubi [A] time = 0.08, antiderivative size = 146, 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 {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^6}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^3 (a+b x) (d+e x)^7} \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) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, 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 )}{(d+e x)^8} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^2}{(d+e x)^8} \, 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)^2}{e^2 (d+e x)^8}-\frac {2 b (b d-a e)}{e^2 (d+e x)^7}+\frac {b^2}{e^2 (d+e x)^6}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x) (d+e x)^7}+\frac {b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^6}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 0.50 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )}{105 e^3 (a+b x) (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.03, 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 = 131, normalized size = 0.90 \begin {gather*} -\frac {21 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 5 \, a b d e + 15 \, a^{2} e^{2} + 7 \, {\left (b^{2} d e + 5 \, a b e^{2}\right )} x}{105 \, {\left (e^{10} x^{7} + 7 \, d e^{9} x^{6} + 21 \, d^{2} e^{8} x^{5} + 35 \, d^{3} e^{7} x^{4} + 35 \, d^{4} e^{6} x^{3} + 21 \, d^{5} e^{5} x^{2} + 7 \, d^{6} e^{4} x + d^{7} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 96, normalized size = 0.66 \begin {gather*} -\frac {{\left (21 \, b^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{2} d x e \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b x e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b d e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{105 \, {\left (x e + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 78, normalized size = 0.53 \begin {gather*} -\frac {\left (21 b^{2} e^{2} x^{2}+35 a b \,e^{2} x +7 b^{2} d e x +15 a^{2} e^{2}+5 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{105 \left (e x +d \right )^{7} \left (b x +a \right ) e^{3}} \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 [B] time = 2.13, size = 77, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (15\,a^2\,e^2+5\,a\,b\,d\,e+35\,a\,b\,e^2\,x+b^2\,d^2+7\,b^2\,d\,e\,x+21\,b^2\,e^2\,x^2\right )}{105\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.43, size = 139, normalized size = 0.95 \begin {gather*} \frac {- 15 a^{2} e^{2} - 5 a b d e - b^{2} d^{2} - 21 b^{2} e^{2} x^{2} + x \left (- 35 a b e^{2} - 7 b^{2} d e\right )}{105 d^{7} e^{3} + 735 d^{6} e^{4} x + 2205 d^{5} e^{5} x^{2} + 3675 d^{4} e^{6} x^{3} + 3675 d^{3} e^{7} x^{4} + 2205 d^{2} e^{8} x^{5} + 735 d e^{9} x^{6} + 105 e^{10} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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