Optimal. Leaf size=254 \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^6}-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^5 (a+b x) (d+e x)^7}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^5 (a+b x) (d+e x)^8}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^9} \]
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Rubi [A] time = 0.131884, antiderivative size = 254, normalized size of antiderivative = 1., 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} \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^6}-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^5 (a+b x) (d+e x)^7}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^5 (a+b x) (d+e x)^8}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^9} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
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)^{10}} \, 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)^{10}} \, 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)^{10}} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{10}}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^9}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^8}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^7}+\frac{b^4}{e^4 (d+e x)^6}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x) (d+e x)^9}+\frac{b (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x) (d+e x)^8}-\frac{6 b^2 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}+\frac{2 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^6}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.0638924, size = 162, normalized size = 0.64 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+35 a^3 b e^3 (d+9 e x)+70 a^4 e^4+5 a b^3 e \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )\right )}{630 e^5 (a+b x) (d+e x)^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 201, normalized size = 0.8 \begin{align*} -{\frac{126\,{x}^{4}{b}^{4}{e}^{4}+420\,{x}^{3}a{b}^{3}{e}^{4}+84\,{x}^{3}{b}^{4}d{e}^{3}+540\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+180\,{x}^{2}a{b}^{3}d{e}^{3}+36\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+315\,x{a}^{3}b{e}^{4}+135\,x{a}^{2}{b}^{2}d{e}^{3}+45\,xa{b}^{3}{d}^{2}{e}^{2}+9\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}+35\,d{e}^{3}{a}^{3}b+15\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+5\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{630\,{e}^{5} \left ( ex+d \right ) ^{9} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61814, size = 568, normalized size = 2.24 \begin{align*} -\frac{126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \,{\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \,{\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14818, size = 356, normalized size = 1.4 \begin{align*} -\frac{{\left (126 \, b^{4} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 84 \, b^{4} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 36 \, b^{4} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 9 \, b^{4} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) + 420 \, a b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 180 \, a b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 45 \, a b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 540 \, a^{2} b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 135 \, a^{2} b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 315 \, a^{3} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 70 \, a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{630 \,{\left (x e + d\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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