Optimal. Leaf size=166 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^4 (a+b x)}+\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x)}-\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^2}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e} \]
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Rubi [A] time = 0.07, antiderivative size = 166, 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 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x)}-\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^4 (a+b x)}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e} \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} \, 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} \, 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^4 (b d-a e)^2}{e^3}-\frac {b^3 (b d-a e) \left (a b+b^2 x\right )}{e^2}+\frac {b^2 \left (a b+b^2 x\right )^2}{e}-\frac {b^3 (b d-a e)^3}{e^3 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {b (b d-a e)^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}-\frac {(b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^2}+\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e}-\frac {(b d-a e)^3 \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 = 0.05, size = 92, normalized size = 0.55 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (b e x \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (b d-a e)^3 \log (d+e x)\right )}{6 e^4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.68, 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} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 115, normalized size = 0.69 \begin {gather*} \frac {2 \, b^{3} e^{3} x^{3} - 3 \, {\left (b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 173, normalized size = 1.04 \begin {gather*} -{\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 )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, b^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, b^{3} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{3} d^{2} x \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 18 \, a b^{2} d x e \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b x e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 149, normalized size = 0.90 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (2 b^{3} e^{3} x^{3}+9 a \,b^{2} e^{3} x^{2}-3 b^{3} d \,e^{2} x^{2}+6 a^{3} e^{3} \ln \left (e x +d \right )-18 a^{2} b d \,e^{2} \ln \left (e x +d \right )+18 a^{2} b \,e^{3} x +18 a \,b^{2} d^{2} e \ln \left (e x +d \right )-18 a \,b^{2} d \,e^{2} x -6 b^{3} d^{3} \ln \left (e x +d \right )+6 b^{3} d^{2} e x \right )}{6 \left (b x +a \right )^{3} 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}}{d+e\,x} \,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}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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