Optimal. Leaf size=183 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) (d+e x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^4 (a+b x)}-\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (2 b d-3 a e)}{e^3 (a+b x)}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)} \]
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Rubi [A] time = 0.09, antiderivative size = 183, 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^2 x \sqrt {a^2+2 a b x+b^2 x^2} (2 b d-3 a e)}{e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) (d+e x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^4 (a+b x)}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2 (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 )^{3/2}}{(d+e x)^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{(d+e x)^2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (2 b d-3 a e)}{e^3}+\frac {b^6 x}{e^2}-\frac {b^3 (b d-a e)^3}{e^3 (d+e x)^2}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {b^2 (2 b d-3 a e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)}+\frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) (d+e x)}+\frac {3 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^4 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 132, normalized size = 0.72 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-2 a^3 e^3+6 a^2 b d e^2+6 a b^2 e \left (-d^2+d e x+e^2 x^2\right )+6 b (d+e x) (b d-a e)^2 \log (d+e x)+b^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )}{2 e^4 (a+b x) (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 2.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 172, normalized size = 0.94 \begin {gather*} \frac {b^{3} e^{3} x^{3} + 2 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} - 3 \, {\left (b^{3} d e^{2} - 2 \, a b^{2} e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2}\right )} x + 6 \, {\left (b^{3} d^{3} - 2 \, a b^{2} d^{2} e + a^{2} b d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 175, normalized size = 0.96 \begin {gather*} 3 \, {\left (b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (b^{3} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, b^{3} d x e \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{2} x e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} + \frac {{\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 216, normalized size = 1.18 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} e^{3} x^{3}+6 a^{2} b \,e^{3} x \ln \left (e x +d \right )-12 a \,b^{2} d \,e^{2} x \ln \left (e x +d \right )+6 a \,b^{2} e^{3} x^{2}+6 b^{3} d^{2} e x \ln \left (e x +d \right )-3 b^{3} d \,e^{2} x^{2}+6 a^{2} b d \,e^{2} \ln \left (e x +d \right )-12 a \,b^{2} d^{2} e \ln \left (e x +d \right )+6 a \,b^{2} d \,e^{2} x +6 b^{3} d^{3} \ln \left (e x +d \right )-4 b^{3} d^{2} e x -2 a^{3} e^{3}+6 a^{2} b d \,e^{2}-6 a \,b^{2} d^{2} e +2 b^{3} d^{3}\right )}{2 \left (b x +a \right )^{3} \left (e x +d \right ) e^{4}} \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 = -0.01 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,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 {3}{2}}}{\left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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