Optimal. Leaf size=205 \[ -\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.385848, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1920, 1951, 12, 1904, 206} \[ -\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1920
Rule 1951
Rule 12
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^6} \, dx &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}+\frac{1}{8} \int \frac{b+2 c x}{x^3 \sqrt{a x^2+b x^3+c x^4}} \, dx\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\int \frac{\frac{1}{2} \left (5 b^2-12 a c\right )+2 b c x}{x^2 \sqrt{a x^2+b x^3+c x^4}} \, dx}{24 a}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}+\frac{\int \frac{\frac{1}{4} b \left (15 b^2-52 a c\right )+\frac{1}{2} c \left (5 b^2-12 a c\right ) x}{x \sqrt{a x^2+b x^3+c x^4}} \, dx}{48 a^2}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}-\frac{\int \frac{3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )}{8 \sqrt{a x^2+b x^3+c x^4}} \, dx}{48 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{128 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x (2 a+b x)}{\sqrt{a x^2+b x^3+c x^4}}\right )}{64 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.225157, size = 160, normalized size = 0.78 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (3 x^4 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (8 a^2 x (b+3 c x)+48 a^3-2 a b x^2 (5 b+26 c x)+15 b^3 x^3\right )\right )}{384 a^{7/2} x^5 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 387, normalized size = 1.9 \begin{align*}{\frac{1}{384\,{x}^{5}{a}^{4}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 48\,{c}^{2}{a}^{5/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}+24\,{c}^{2}\sqrt{c{x}^{2}+bx+a}{x}^{5}ab-72\,c{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{2}-48\,{c}^{2}\sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{2}-30\,c\sqrt{c{x}^{2}+bx+a}{x}^{5}{b}^{3}-24\,c \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}ab+84\,c\sqrt{c{x}^{2}+bx+a}{x}^{4}a{b}^{2}+15\,\sqrt{a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{4}+48\,c \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}{a}^{2}+30\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}{b}^{3}-30\,\sqrt{c{x}^{2}+bx+a}{x}^{4}{b}^{4}-60\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}a{b}^{2}+80\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}x{a}^{2}b-96\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \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 \frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.12879, size = 757, normalized size = 3.69 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{a} x^{5} \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 4 \,{\left (8 \, a^{3} b x + 48 \, a^{4} +{\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{768 \, a^{4} x^{5}}, -\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-a} x^{5} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \,{\left (8 \, a^{3} b x + 48 \, a^{4} +{\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{384 \, a^{4} x^{5}}\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 \frac{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]