Optimal. Leaf size=239 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) (d+e x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^5 (a+b x)}+\frac {6 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^5 (a+b x)}-\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^5 (a+b x)} \]
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Rubi [A] time = 0.16, antiderivative size = 239, 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^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^5 (a+b x)}-\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^5 (a+b x)}+\frac {6 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) (d+e x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^5 (a+b x)} \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) \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 {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^2} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {6 b^2 (b d-a e)^2}{e^4}+\frac {(-b d+a e)^4}{e^4 (d+e x)^2}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)}-\frac {4 b^3 (b d-a e) (d+e x)}{e^4}+\frac {b^4 (d+e x)^2}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {6 b^2 (b d-a e)^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}-\frac {2 b^3 (b d-a e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac {b^4 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 183, normalized size = 0.77 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )-12 b (d+e x) (b d-a e)^3 \log (d+e x)+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )}{3 e^5 (a+b x) (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 2.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \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 = 267, normalized size = 1.12 \begin {gather*} \frac {b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 2 \, {\left (b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (3 \, b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3}\right )} x - 12 \, {\left (b^{4} d^{4} - 3 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} - a^{3} b d e^{3} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 268, normalized size = 1.12 \begin {gather*} -4 \, {\left (b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{3} \, {\left (b^{4} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, b^{4} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 9 \, b^{4} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{3} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 24 \, a b^{3} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b^{2} x e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} - \frac {{\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\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 = 327, normalized size = 1.37 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{4} e^{4} x^{4}+6 a \,b^{3} e^{4} x^{3}-2 b^{4} d \,e^{3} x^{3}+12 a^{3} b \,e^{4} x \ln \left (e x +d \right )-36 a^{2} b^{2} d \,e^{3} x \ln \left (e x +d \right )+18 a^{2} b^{2} e^{4} x^{2}+36 a \,b^{3} d^{2} e^{2} x \ln \left (e x +d \right )-18 a \,b^{3} d \,e^{3} x^{2}-12 b^{4} d^{3} e x \ln \left (e x +d \right )+6 b^{4} d^{2} e^{2} x^{2}+12 a^{3} b d \,e^{3} \ln \left (e x +d \right )-36 a^{2} b^{2} d^{2} e^{2} \ln \left (e x +d \right )+18 a^{2} b^{2} d \,e^{3} x +36 a \,b^{3} d^{3} e \ln \left (e x +d \right )-24 a \,b^{3} d^{2} e^{2} x -12 b^{4} d^{4} \ln \left (e x +d \right )+9 b^{4} d^{3} e x -3 a^{4} e^{4}+12 a^{3} b d \,e^{3}-18 a^{2} b^{2} d^{2} e^{2}+12 a \,b^{3} d^{3} e -3 b^{4} d^{4}\right )}{3 \left (b x +a \right )^{3} \left (e x +d \right ) e^{5}} \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 = -0.00 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\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 (a + b x\right ) \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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