Optimal. Leaf size=39 \[ \frac{2 \left (a+b x+c x^2\right )^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0164574, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {682} \[ \frac{2 \left (a+b x+c x^2\right )^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 682
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{5/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}\\ \end{align*}
Mathematica [A] time = 0.025451, size = 38, normalized size = 0.97 \[ \frac{2 (a+x (b+c x))^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 38, normalized size = 1. \begin{align*} -{\frac{2}{5\, \left ( 2\,cx+b \right ) ^{5}{d}^{6} \left ( 4\,ac-{b}^{2} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 24.2278, size = 383, normalized size = 9.82 \begin{align*} \frac{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{5 \,{\left (32 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{6} x^{5} + 80 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{6} x^{4} + 80 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{6} x^{3} + 40 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{6} x^{2} + 10 \,{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{6} x +{\left (b^{7} - 4 \, a b^{5} c\right )} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.84344, size = 802, normalized size = 20.56 \begin{align*} \frac{80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{8} c^{\frac{9}{2}} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{7} b c^{4} + 560 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{6} b^{2} c^{\frac{7}{2}} + 560 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} b^{3} c^{3} + 360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} b^{4} c^{\frac{5}{2}} - 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a b^{2} c^{\frac{7}{2}} + 160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a^{2} c^{\frac{9}{2}} + 160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} b^{5} c^{2} - 160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a b^{3} c^{3} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a^{2} b c^{4} + 50 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} b^{6} c^{\frac{3}{2}} - 120 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a b^{4} c^{\frac{5}{2}} + 240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{2} b^{2} c^{\frac{7}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b^{7} c - 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a b^{5} c^{2} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{2} b^{3} c^{3} + b^{8} \sqrt{c} - 6 \, a b^{6} c^{\frac{3}{2}} + 16 \, a^{2} b^{4} c^{\frac{5}{2}} - 16 \, a^{3} b^{2} c^{\frac{7}{2}} + 16 \, a^{4} c^{\frac{9}{2}}}{80 \,{\left (2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b \sqrt{c} + b^{2} - 2 \, a c\right )}^{5} c^{3} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]