Optimal. Leaf size=212 \[ -\frac{2 (f+g x)^{5/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5}-\frac{2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac{2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.339492, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {897, 1153} \[ -\frac{2 (f+g x)^{5/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5}-\frac{2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac{2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 897
Rule 1153
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt{f+g x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^2 \left (\frac{c f^2-b f g+a g^2}{g^2}-\frac{(2 c f-b g) x^2}{g^2}+\frac{c x^4}{g^2}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{(-e f+d g)^2 \left (c f^2-b f g+a g^2\right )}{g^4}+\frac{(e f-d g) (-2 c f (2 e f-d g)+g (3 b e f-b d g-2 a e g)) x^2}{g^4}+\frac{\left (-e g (3 b e f-2 b d g-a e g)+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) x^4}{g^4}+\frac{e (-4 c e f+2 c d g+b e g) x^6}{g^4}+\frac{c e^2 x^8}{g^4}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}{g^5}-\frac{2 (e f-d g) (2 c f (2 e f-d g)-g (3 b e f-b d g-2 a e g)) (f+g x)^{3/2}}{3 g^5}-\frac{2 \left (e g (3 b e f-2 b d g-a e g)-c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^5}-\frac{2 e (4 c e f-2 c d g-b e g) (f+g x)^{7/2}}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5}\\ \end{align*}
Mathematica [A] time = 0.361759, size = 184, normalized size = 0.87 \[ \frac{2 \sqrt{f+g x} \left (-63 (f+g x)^2 \left (-e g (a e g+2 b d g-3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )+315 (e f-d g)^2 \left (g (a g-b f)+c f^2\right )-105 (f+g x) (e f-d g) (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g))-45 e (f+g x)^3 (-b e g-2 c d g+4 c e f)+35 c e^2 (f+g x)^4\right )}{315 g^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.059, size = 315, normalized size = 1.5 \begin{align*}{\frac{70\,c{e}^{2}{x}^{4}{g}^{4}+90\,b{e}^{2}{g}^{4}{x}^{3}+180\,cde{g}^{4}{x}^{3}-80\,c{e}^{2}f{g}^{3}{x}^{3}+126\,a{e}^{2}{g}^{4}{x}^{2}+252\,bde{g}^{4}{x}^{2}-108\,b{e}^{2}f{g}^{3}{x}^{2}+126\,c{d}^{2}{g}^{4}{x}^{2}-216\,cdef{g}^{3}{x}^{2}+96\,c{e}^{2}{f}^{2}{g}^{2}{x}^{2}+420\,ade{g}^{4}x-168\,a{e}^{2}f{g}^{3}x+210\,b{d}^{2}{g}^{4}x-336\,bdef{g}^{3}x+144\,b{e}^{2}{f}^{2}{g}^{2}x-168\,c{d}^{2}f{g}^{3}x+288\,cde{f}^{2}{g}^{2}x-128\,c{e}^{2}{f}^{3}gx+630\,a{d}^{2}{g}^{4}-840\,adef{g}^{3}+336\,a{e}^{2}{f}^{2}{g}^{2}-420\,b{d}^{2}f{g}^{3}+672\,bde{f}^{2}{g}^{2}-288\,b{e}^{2}{f}^{3}g+336\,c{d}^{2}{f}^{2}{g}^{2}-576\,cde{f}^{3}g+256\,c{e}^{2}{f}^{4}}{315\,{g}^{5}}\sqrt{gx+f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.982756, size = 352, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (35 \,{\left (g x + f\right )}^{\frac{9}{2}} c e^{2} - 45 \,{\left (4 \, c e^{2} f -{\left (2 \, c d e + b e^{2}\right )} g\right )}{\left (g x + f\right )}^{\frac{7}{2}} + 63 \,{\left (6 \, c e^{2} f^{2} - 3 \,{\left (2 \, c d e + b e^{2}\right )} f g +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{5}{2}} - 105 \,{\left (4 \, c e^{2} f^{3} - 3 \,{\left (2 \, c d e + b e^{2}\right )} f^{2} g + 2 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{2} -{\left (b d^{2} + 2 \, a d e\right )} g^{3}\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 315 \,{\left (c e^{2} f^{4} + a d^{2} g^{4} -{\left (2 \, c d e + b e^{2}\right )} f^{3} g +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} -{\left (b d^{2} + 2 \, a d e\right )} f g^{3}\right )} \sqrt{g x + f}\right )}}{315 \, g^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.76131, size = 586, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (35 \, c e^{2} g^{4} x^{4} + 128 \, c e^{2} f^{4} + 315 \, a d^{2} g^{4} - 144 \,{\left (2 \, c d e + b e^{2}\right )} f^{3} g + 168 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} - 210 \,{\left (b d^{2} + 2 \, a d e\right )} f g^{3} - 5 \,{\left (8 \, c e^{2} f g^{3} - 9 \,{\left (2 \, c d e + b e^{2}\right )} g^{4}\right )} x^{3} + 3 \,{\left (16 \, c e^{2} f^{2} g^{2} - 18 \,{\left (2 \, c d e + b e^{2}\right )} f g^{3} + 21 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{4}\right )} x^{2} -{\left (64 \, c e^{2} f^{3} g - 72 \,{\left (2 \, c d e + b e^{2}\right )} f^{2} g^{2} + 84 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{3} - 105 \,{\left (b d^{2} + 2 \, a d e\right )} g^{4}\right )} x\right )} \sqrt{g x + f}}{315 \, g^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 96.045, size = 1001, normalized size = 4.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12253, size = 490, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{g x + f} a d^{2} + \frac{105 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} b d^{2}}{g} + \frac{210 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} a d e}{g} + \frac{21 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} c d^{2}}{g^{2}} + \frac{42 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} b d e}{g^{2}} + \frac{21 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} a e^{2}}{g^{2}} + \frac{18 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} - 35 \, \sqrt{g x + f} f^{3}\right )} c d e}{g^{3}} + \frac{9 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} - 35 \, \sqrt{g x + f} f^{3}\right )} b e^{2}}{g^{3}} + \frac{{\left (35 \,{\left (g x + f\right )}^{\frac{9}{2}} - 180 \,{\left (g x + f\right )}^{\frac{7}{2}} f + 378 \,{\left (g x + f\right )}^{\frac{5}{2}} f^{2} - 420 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{3} + 315 \, \sqrt{g x + f} f^{4}\right )} c e^{2}}{g^{4}}\right )}}{315 \, g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]