Optimal. Leaf size=257 \[ -\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac{x \left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}+\frac{b x \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.588441, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1919, 1949, 12, 1914, 621, 206} \[ -\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac{x \left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}+\frac{b x \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1919
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^2 \sqrt{a x^2+b x^3+c x^4} \, dx &=\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}+\frac{\int \frac{x^3 \left (-3 a b-\frac{1}{2} \left (7 b^2-16 a c\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{40 c}\\ &=-\frac{\left (7 b^2-16 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}-\frac{\int \frac{x^2 \left (-a \left (7 b^2-16 a c\right )-\frac{1}{4} b \left (35 b^2-116 a c\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{120 c^2}\\ &=\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}-\frac{\left (7 b^2-16 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}+\frac{\int \frac{x \left (-\frac{1}{4} a b \left (35 b^2-116 a c\right )-\frac{1}{8} \left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{240 c^3}\\ &=\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac{\left (7 b^2-16 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}-\frac{\int -\frac{15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x}{16 \sqrt{a x^2+b x^3+c x^4}} \, dx}{240 c^4}\\ &=\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac{\left (7 b^2-16 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{256 c^4}\\ &=\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac{\left (7 b^2-16 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^4 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac{\left (7 b^2-16 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x \sqrt{a+b x+c x^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 )}{128 c^4 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac{\left (7 b^2-16 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}{240 c^2}+\frac{x^2 (b+8 c x) \sqrt{a x^2+b x^3+c x^4}}{40 c}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.241455, size = 180, normalized size = 0.7 \[ \frac{2 \sqrt{c} x (a+x (b+c x)) \left (128 c^2 \left (-2 a^2+a c x^2+3 c^2 x^4\right )+4 b^2 c \left (115 a-14 c x^2\right )+8 b c^2 x \left (6 c x^2-29 a\right )+70 b^3 c x-105 b^4\right )+15 x \left (48 a^2 b c^2-40 a b^3 c+7 b^5\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3840 c^{9/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 310, normalized size = 1.2 \begin{align*}{\frac{1}{3840\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 768\,{x}^{2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{9/2}-672\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}xb-512\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}a+560\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{b}^{2}+720\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}xab-420\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}x{b}^{3}+360\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}a{b}^{2}-210\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{b}^{4}+720\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}b{c}^{3}-600\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{3}{c}^{2}+105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{5}c \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c x^{4} + b x^{3} + a x^{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73634, size = 898, normalized size = 3.49 \begin{align*} \left [\frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \,{\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \,{\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{7680 \, c^{5} x}, -\frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \,{\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \,{\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{3840 \, c^{5} x}\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*} \int x^{2} \sqrt{x^{2} \left (a + b x + c x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18191, size = 382, normalized size = 1.49 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x \mathrm{sgn}\left (x\right ) + \frac{b \mathrm{sgn}\left (x\right )}{c}\right )} x - \frac{7 \, b^{2} c^{2} \mathrm{sgn}\left (x\right ) - 16 \, a c^{3} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} x + \frac{35 \, b^{3} c \mathrm{sgn}\left (x\right ) - 116 \, a b c^{2} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} x - \frac{105 \, b^{4} \mathrm{sgn}\left (x\right ) - 460 \, a b^{2} c \mathrm{sgn}\left (x\right ) + 256 \, a^{2} c^{2} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} - \frac{{\left (7 \, b^{5} \mathrm{sgn}\left (x\right ) - 40 \, a b^{3} c \mathrm{sgn}\left (x\right ) + 48 \, a^{2} b c^{2} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} + \frac{{\left (105 \, b^{5} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 600 \, a b^{3} c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 720 \, a^{2} b c^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 210 \, \sqrt{a} b^{4} \sqrt{c} - 920 \, a^{\frac{3}{2}} b^{2} c^{\frac{3}{2}} + 512 \, a^{\frac{5}{2}} c^{\frac{5}{2}}\right )} \mathrm{sgn}\left (x\right )}{3840 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]