Optimal. Leaf size=258 \[ \frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^5 (a+b x)}+\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) \sqrt{d+e x}} \]
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Rubi [A] time = 0.105694, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 43} \[ \frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^5 (a+b x)}+\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) \sqrt{d+e x}} \]
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)^{3/2}} \, 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)^{3/2}} \, 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)^{3/2}} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{3/2}}-\frac{4 b (b d-a e)^3}{e^4 \sqrt{d+e x}}+\frac{6 b^2 (b d-a e)^2 \sqrt{d+e x}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{3/2}}{e^4}+\frac{b^4 (d+e x)^{5/2}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt{d+e x}}-\frac{8 b (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac{4 b^2 (b d-a e)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac{8 b^3 (b d-a e) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}+\frac{2 b^4 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0998249, size = 169, normalized size = 0.66 \[ \frac{2 \sqrt{(a+b x)^2} \left (70 a^2 b^2 e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+140 a^3 b e^3 (2 d+e x)-35 a^4 e^4+28 a b^3 e \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+b^4 \left (16 d^2 e^2 x^2-64 d^3 e x-128 d^4-8 d e^3 x^3+5 e^4 x^4\right )\right )}{35 e^5 (a+b x) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 202, normalized size = 0.8 \begin{align*} -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-56\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+112\,{x}^{2}a{b}^{3}d{e}^{3}-32\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-280\,x{a}^{3}b{e}^{4}+560\,x{a}^{2}{b}^{2}d{e}^{3}-448\,xa{b}^{3}{d}^{2}{e}^{2}+128\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}-560\,d{e}^{3}{a}^{3}b+1120\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-896\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{35\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15846, size = 381, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} -{\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} a}{5 \, \sqrt{e x + d} e^{4}} + \frac{2 \,{\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 336 \, a b^{2} d^{3} e - 280 \, a^{2} b d^{2} e^{2} + 70 \, a^{3} d e^{3} -{\left (8 \, b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} +{\left (16 \, b^{3} d^{2} e^{2} - 42 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} -{\left (64 \, b^{3} d^{3} e - 168 \, a b^{2} d^{2} e^{2} + 140 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} b}{35 \, \sqrt{e x + d} e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.988529, size = 412, normalized size = 1.6 \begin{align*} \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \,{\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20923, size = 441, normalized size = 1.71 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} e^{30} \mathrm{sgn}\left (b x + a\right ) - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d e^{30} \mathrm{sgn}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{2} e^{30} \mathrm{sgn}\left (b x + a\right ) - 140 \, \sqrt{x e + d} b^{4} d^{3} e^{30} \mathrm{sgn}\left (b x + a\right ) + 28 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} e^{31} \mathrm{sgn}\left (b x + a\right ) - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d e^{31} \mathrm{sgn}\left (b x + a\right ) + 420 \, \sqrt{x e + d} a b^{3} d^{2} e^{31} \mathrm{sgn}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} e^{32} \mathrm{sgn}\left (b x + a\right ) - 420 \, \sqrt{x e + d} a^{2} b^{2} d e^{32} \mathrm{sgn}\left (b x + a\right ) + 140 \, \sqrt{x e + d} a^{3} b e^{33} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-35\right )} - \frac{2 \,{\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 )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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