Optimal. Leaf size=444 \[ -\frac{2 \left (-2 c x \left (-c^2 \left (16 a^2 f^2+12 a b e f+b^2 \left (-\left (2 d f+e^2\right )\right )\right )+b^2 c f (14 a f+b e)-c^3 \left (8 b d e-4 a \left (2 d f+e^2\right )\right )-2 b^4 f^2+8 c^4 d^2\right )-4 b c^2 \left (8 a^2 f^2+a c \left (2 d f+e^2\right )+2 c^2 d^2\right )+48 a^2 c^3 e f+4 b^2 c^2 e (2 c d-3 a f)+b^3 c \left (10 a f^2-c \left (2 d f+e^2\right )\right )+2 b^4 c e f+b^5 \left (-f^2\right )\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )+2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{f^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.450728, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1660, 12, 621, 206} \[ -\frac{2 \left (-2 c x \left (-c^2 \left (16 a^2 f^2+12 a b e f+b^2 \left (-\left (2 d f+e^2\right )\right )\right )+b^2 c f (14 a f+b e)-c^3 \left (8 b d e-4 a \left (2 d f+e^2\right )\right )-2 b^4 f^2+8 c^4 d^2\right )-4 b c^2 \left (8 a^2 f^2+a c \left (2 d f+e^2\right )+2 c^2 d^2\right )+48 a^2 c^3 e f+4 b^2 c^2 e (2 c d-3 a f)+b^3 c \left (10 a f^2-c \left (2 d f+e^2\right )\right )+2 b^4 c e f+b^5 \left (-f^2\right )\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )+2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{f^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1660
Rule 12
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (d+e x+f x^2\right )^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{8 c^4 d^2+b^4 f^2-b^2 c f (2 b e+a f)-c^3 \left (8 b d e-4 a \left (e^2+2 d f\right )\right )-c^2 \left (4 a^2 f^2-b^2 \left (e^2+2 d f\right )\right )}{2 c^3}-\frac{3 \left (b^2-4 a c\right ) f (2 c e-b f) x}{2 c^2}+\frac{3}{2} \left (4 a-\frac{b^2}{c}\right ) f^2 x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 b^4 c e f+48 a^2 c^3 e f-b^5 f^2+4 b^2 c^2 e (2 c d-3 a f)+b^3 c \left (10 a f^2-c \left (e^2+2 d f\right )\right )-4 b c^2 \left (2 c^2 d^2+8 a^2 f^2+a c \left (e^2+2 d f\right )\right )-2 c \left (8 c^4 d^2-2 b^4 f^2+b^2 c f (b e+14 a f)-c^3 \left (8 b d e-4 a \left (e^2+2 d f\right )\right )-c^2 \left (12 a b e f+16 a^2 f^2-b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{4 \int \frac{3 \left (b^2-4 a c\right )^2 f^2}{4 c^2 \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 b^4 c e f+48 a^2 c^3 e f-b^5 f^2+4 b^2 c^2 e (2 c d-3 a f)+b^3 c \left (10 a f^2-c \left (e^2+2 d f\right )\right )-4 b c^2 \left (2 c^2 d^2+8 a^2 f^2+a c \left (e^2+2 d f\right )\right )-2 c \left (8 c^4 d^2-2 b^4 f^2+b^2 c f (b e+14 a f)-c^3 \left (8 b d e-4 a \left (e^2+2 d f\right )\right )-c^2 \left (12 a b e f+16 a^2 f^2-b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{f^2 \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c^2}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 b^4 c e f+48 a^2 c^3 e f-b^5 f^2+4 b^2 c^2 e (2 c d-3 a f)+b^3 c \left (10 a f^2-c \left (e^2+2 d f\right )\right )-4 b c^2 \left (2 c^2 d^2+8 a^2 f^2+a c \left (e^2+2 d f\right )\right )-2 c \left (8 c^4 d^2-2 b^4 f^2+b^2 c f (b e+14 a f)-c^3 \left (8 b d e-4 a \left (e^2+2 d f\right )\right )-c^2 \left (12 a b e f+16 a^2 f^2-b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{c^2}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 b^4 c e f+48 a^2 c^3 e f-b^5 f^2+4 b^2 c^2 e (2 c d-3 a f)+b^3 c \left (10 a f^2-c \left (e^2+2 d f\right )\right )-4 b c^2 \left (2 c^2 d^2+8 a^2 f^2+a c \left (e^2+2 d f\right )\right )-2 c \left (8 c^4 d^2-2 b^4 f^2+b^2 c f (b e+14 a f)-c^3 \left (8 b d e-4 a \left (e^2+2 d f\right )\right )-c^2 \left (12 a b e f+16 a^2 f^2-b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{f^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.31537, size = 387, normalized size = 0.87 \[ \frac{2 \left (2 b^2 c \left (21 a^2 f^2 x-2 a c \left (d (e-6 f x)+x \left (-3 e^2+3 e f x-7 f^2 x^2\right )\right )+c^2 x \left (3 d^2+2 d x (f x-6 e)+e^2 x^2\right )\right )+b^3 \left (-3 a^2 f^2+18 a c f^2 x^2+c^2 \left (-d^2+6 d x (f x-e)+e x^2 (3 e+2 f x)\right )\right )+4 b c \left (2 a^2 c \left (2 d f+e^2-6 e f x\right )+5 a^3 f^2+3 a c^2 (d-e x) (d+x (2 f x-e))+2 c^3 d x^2 (3 d-2 e x)\right )+8 c^2 \left (-2 a^2 c \left (d e+f x^2 (3 e+2 f x)\right )+a^3 (-f) (4 e+3 f x)+a c^2 x \left (3 d^2+2 d f x^2+e^2 x^2\right )+2 c^3 d^2 x^3\right )-2 b^4 f^2 x \left (3 a+2 c x^2\right )-3 b^5 f^2 x^2\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac{f^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.061, size = 1786, normalized size = 4. \begin{align*} \text{result too large to display} \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 [A] time = 27.422, size = 3252, normalized size = 7.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.38829, size = 792, normalized size = 1.78 \begin{align*} -\frac{f^{2} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} + \frac{2 \,{\left ({\left ({\left (\frac{2 \,{\left (8 \, c^{5} d^{2} + 2 \, b^{2} c^{3} d f + 8 \, a c^{4} d f - 2 \, b^{4} c f^{2} + 14 \, a b^{2} c^{2} f^{2} - 16 \, a^{2} c^{3} f^{2} - 8 \, b c^{4} d e + b^{3} c^{2} f e - 12 \, a b c^{3} f e + b^{2} c^{3} e^{2} + 4 \, a c^{4} e^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{4} d^{2} + 2 \, b^{3} c^{2} d f + 8 \, a b c^{3} d f - b^{5} f^{2} + 6 \, a b^{3} c f^{2} - 8 \, b^{2} c^{3} d e - 4 \, a b^{2} c^{2} f e - 16 \, a^{2} c^{3} f e + b^{3} c^{2} e^{2} + 4 \, a b c^{3} e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (b^{2} c^{3} d^{2} + 4 \, a c^{4} d^{2} + 4 \, a b^{2} c^{2} d f - a b^{4} f^{2} + 7 \, a^{2} b^{2} c f^{2} - 4 \, a^{3} c^{2} f^{2} - b^{3} c^{2} d e - 4 \, a b c^{3} d e - 8 \, a^{2} b c^{2} f e + 2 \, a b^{2} c^{2} e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} c^{2} d^{2} - 12 \, a b c^{3} d^{2} - 16 \, a^{2} b c^{2} d f + 3 \, a^{2} b^{3} f^{2} - 20 \, a^{3} b c f^{2} + 4 \, a b^{2} c^{2} d e + 16 \, a^{2} c^{3} d e + 32 \, a^{3} c^{2} f e - 8 \, a^{2} b c^{2} e^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]