Optimal. Leaf size=146 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \]
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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}}{4 e^3 (a+b x) (d+e x)^4}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \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)^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 )}{(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 \frac {(a+b x)^2}{(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)^2}{e^2 (d+e x)^7}-\frac {2 b (b d-a e)}{e^2 (d+e x)^6}+\frac {b^2}{e^2 (d+e x)^5}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x) (d+e x)^6}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}\\ \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 (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \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 = 0.40, size = 120, normalized size = 0.82 \begin {gather*} -\frac {15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \, {\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 96, normalized size = 0.66 \begin {gather*} -\frac {{\left (15 \, b^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{2} d x e \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 24 \, a b x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b d e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\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 = 0.05, size = 78, normalized size = 0.53 \begin {gather*} -\frac {\left (15 b^{2} e^{2} x^{2}+24 a b \,e^{2} x +6 b^{2} d e x +10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 \left (e x +d \right )^{6} \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 = 2.12, size = 77, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+24\,a\,b\,e^2\,x+b^2\,d^2+6\,b^2\,d\,e\,x+15\,b^2\,e^2\,x^2\right )}{60\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 128, normalized size = 0.88 \begin {gather*} \frac {- 10 a^{2} e^{2} - 4 a b d e - b^{2} d^{2} - 15 b^{2} e^{2} x^{2} + x \left (- 24 a b e^{2} - 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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