Optimal. Leaf size=41 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (d+e x)^3 (b d-a e)} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {767} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (d+e x)^3 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 767
Rubi steps
\begin {align*} \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx &=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e) (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{3 e^3 (a+b x) (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 84, normalized size = 2.05 \begin {gather*} -\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 94, normalized size = 2.29 \begin {gather*} -\frac {{\left (3 \, b^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, b^{2} d x e \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b x e^{2} \mathrm {sgn}\left (b x + a\right ) + a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 76, normalized size = 1.85 \begin {gather*} -\frac {\left (3 b^{2} e^{2} x^{2}+3 a b \,e^{2} x +3 b^{2} d e x +a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (e x +d \right )^{3} \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 = 2.10, size = 75, normalized size = 1.83 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a^2\,e^2+a\,b\,d\,e+3\,a\,b\,e^2\,x+b^2\,d^2+3\,b^2\,d\,e\,x+3\,b^2\,e^2\,x^2\right )}{3\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.65, size = 88, normalized size = 2.15 \begin {gather*} \frac {- a^{2} e^{2} - a b d e - b^{2} d^{2} - 3 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} - 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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