Optimal. Leaf size=186 \[ -\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4 (a+b x) (d+e x)^2}-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^4 (a+b x)}+\frac {b^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \]
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Rubi [A] time = 0.09, antiderivative size = 186, 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} \[ -\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4 (a+b x) (d+e x)^2}-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^4 (a+b x)}+\frac {b^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \]
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)^3} \, 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)^3} \, 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^6}{e^3}-\frac {b^3 (b d-a e)^3}{e^3 (d+e x)^3}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^2}-\frac {3 b^5 (b d-a e)}{e^3 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {b^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x) (d+e x)^2}-\frac {3 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) (d+e x)}-\frac {3 b^2 (b d-a e) \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 = 0.06, size = 131, normalized size = 0.70 \[ -\frac {\sqrt {(a+b x)^2} \left (a^3 e^3+3 a^2 b e^2 (d+2 e x)-3 a b^2 d e (3 d+4 e x)+6 b^2 (d+e x)^2 (b d-a e) \log (d+e x)+b^3 \left (5 d^3+4 d^2 e x-4 d e^2 x^2-2 e^3 x^3\right )\right )}{2 e^4 (a+b x) (d+e x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 188, normalized size = 1.01 \[ \frac {2 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d e^{2} x^{2} - 5 \, b^{3} d^{3} + 9 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 2 \, {\left (2 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} d^{3} - a b^{2} d^{2} e + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (b^{3} d^{2} e - a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 170, normalized size = 0.91 \[ b^{3} x e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (b^{3} d \mathrm {sgn}\left (b x + a\right ) - a b^{2} e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (5 \, b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 9 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-4\right )}}{2 \, {\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 219, normalized size = 1.18 \[ \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (6 a \,b^{2} e^{3} x^{2} \ln \left (e x +d \right )-6 b^{3} d \,e^{2} x^{2} \ln \left (e x +d \right )+2 b^{3} e^{3} x^{3}+12 a \,b^{2} d \,e^{2} x \ln \left (e x +d \right )-12 b^{3} d^{2} e x \ln \left (e x +d \right )+4 b^{3} d \,e^{2} x^{2}-6 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e \ln \left (e x +d \right )+12 a \,b^{2} d \,e^{2} x -6 b^{3} d^{3} \ln \left (e x +d \right )-4 b^{3} d^{2} e x -a^{3} e^{3}-3 a^{2} b d \,e^{2}+9 a \,b^{2} d^{2} e -5 b^{3} d^{3}\right )}{2 \left (b x +a \right )^{3} \left (e x +d \right )^{2} e^{4}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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