Optimal. Leaf size=149 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{105 (d+e x)^5 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{21 (d+e x)^6 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{7 (d+e x)^7 (b d-a e)} \]
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Rubi [A] time = 0.0655807, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {770, 21, 45, 37} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{105 (d+e x)^5 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{21 (d+e x)^6 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{7 (d+e x)^7 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 45
Rule 37
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)^8} \, 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)^8} \, 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)^8} \, dx}{a b+b^2 x}\\ &=\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (b d-a e) (d+e x)^7}+\frac{\left (2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^4}{(d+e x)^7} \, dx}{7 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (b d-a e) (d+e x)^7}+\frac{b (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{21 (b d-a e)^2 (d+e x)^6}+\frac{\left (b^3 \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^4}{(d+e x)^6} \, dx}{21 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (b d-a e) (d+e x)^7}+\frac{b (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{21 (b d-a e)^2 (d+e x)^6}+\frac{b^2 (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{105 (b d-a e)^3 (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.0607231, size = 162, normalized size = 1.09 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+10 a^3 b e^3 (d+7 e x)+15 a^4 e^4+3 a b^3 e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )}{105 e^5 (a+b x) (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 201, normalized size = 1.4 \begin{align*} -{\frac{35\,{x}^{4}{b}^{4}{e}^{4}+105\,{x}^{3}a{b}^{3}{e}^{4}+35\,{x}^{3}{b}^{4}d{e}^{3}+126\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+63\,{x}^{2}a{b}^{3}d{e}^{3}+21\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+70\,x{a}^{3}b{e}^{4}+42\,x{a}^{2}{b}^{2}d{e}^{3}+21\,xa{b}^{3}{d}^{2}{e}^{2}+7\,x{b}^{4}{d}^{3}e+15\,{a}^{4}{e}^{4}+10\,d{e}^{3}{a}^{3}b+6\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+3\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{105\,{e}^{5} \left ( ex+d \right ) ^{7} \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 [B] time = 1.65026, size = 512, normalized size = 3.44 \begin{align*} -\frac{35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \,{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \,{\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} 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 [B] time = 1.15466, size = 356, normalized size = 2.39 \begin{align*} -\frac{{\left (35 \, b^{4} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 35 \, b^{4} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 21 \, b^{4} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 7 \, b^{4} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) + 105 \, a b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 63 \, a b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 21 \, a b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 126 \, a^{2} b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 42 \, a^{2} b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 70 \, a^{3} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{105 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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