Optimal. Leaf size=162 \[ \frac{2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^5}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0700174, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {698} \[ \frac{2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^5}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 \sqrt{d+e x}}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt{d+e x}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{3/2}}{e^4}+\frac{c^2 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^2}{e^5 \sqrt{d+e x}}-\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}{e^5}+\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{3 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}\\ \end{align*}
Mathematica [A] time = 0.134387, size = 171, normalized size = 1.06 \[ \frac{-70 e^2 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )+28 c e \left (5 a e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 b \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )-6 c^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )}{105 e^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 194, normalized size = 1.2 \begin{align*} -{\frac{-30\,{c}^{2}{x}^{4}{e}^{4}-84\,bc{e}^{4}{x}^{3}+48\,{c}^{2}d{e}^{3}{x}^{3}-140\,ac{e}^{4}{x}^{2}-70\,{b}^{2}{e}^{4}{x}^{2}+168\,bcd{e}^{3}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-420\,ab{e}^{4}x+560\,acd{e}^{3}x+280\,{b}^{2}d{e}^{3}x-672\,bc{d}^{2}{e}^{2}x+384\,{c}^{2}{d}^{3}ex+210\,{a}^{2}{e}^{4}-840\,abd{e}^{3}+1120\,ac{d}^{2}{e}^{2}+560\,{b}^{2}{d}^{2}{e}^{2}-1344\,bc{d}^{3}e+768\,{c}^{2}{d}^{4}}{105\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.971692, size = 248, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} - 42 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 210 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{105 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{105 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.32187, size = 425, normalized size = 2.62 \begin{align*} \frac{2 \,{\left (15 \, c^{2} e^{4} x^{4} - 384 \, c^{2} d^{4} + 672 \, b c d^{3} e + 420 \, a b d e^{3} - 105 \, a^{2} e^{4} - 280 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 6 \,{\left (4 \, c^{2} d e^{3} - 7 \, b c e^{4}\right )} x^{3} +{\left (48 \, c^{2} d^{2} e^{2} - 84 \, b c d e^{3} + 35 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \,{\left (96 \, c^{2} d^{3} e - 168 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 70 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 29.9527, size = 182, normalized size = 1.12 \begin{align*} \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (4 b c e - 8 c^{2} d\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (4 a c e^{2} + 2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (4 a b e^{3} - 8 a c d e^{2} - 4 b^{2} d e^{2} + 12 b c d^{2} e - 8 c^{2} d^{3}\right )}{e^{5}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{5} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11763, size = 342, normalized size = 2.11 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} e^{30} - 84 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d e^{30} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt{x e + d} c^{2} d^{3} e^{30} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} b c e^{31} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} b c d e^{31} + 630 \, \sqrt{x e + d} b c d^{2} e^{31} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{32} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c e^{32} - 210 \, \sqrt{x e + d} b^{2} d e^{32} - 420 \, \sqrt{x e + d} a c d e^{32} + 210 \, \sqrt{x e + d} a b e^{33}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]