Integrand size = 40, antiderivative size = 1384 \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:
2*(A*e^2+C*d^2)*(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g)*(b-(-4*a*c+b^2)^(1/2)+2*c *x)*((-d*g+e*f)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f-(b+(-4*a*c+b^2)^(1/2)) *g)/(e*x+d))^(1/2)*EllipticE((2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)*(g*x+f )^(1/2)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(e*x+d)^(1/2),((2*c*d-(b+(- 4*a*c+b^2)^(1/2))*e)*(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)/(2*c*d-(b-(-4*a*c+b^ 2)^(1/2))*e)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g))^(1/2))/e/(2*c*d-(b-(-4*a*c+ b^2)^(1/2))*e)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b-(-4*a*c+b^ 2)^(1/2))*g)^(1/2)/(-d*g+e*f)/((-d*g+e*f)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2* c*f-(b-(-4*a*c+b^2)^(1/2))*g)/(e*x+d))^(1/2)/(c*x^2+b*x+a)^(1/2)-4*((b+(-4 *a*c+b^2)^(1/2))*C*d*e-c*(-A*e^2+C*d^2))*(b-(-4*a*c+b^2)^(1/2)+2*c*x)*((-d *g+e*f)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g)/(e*x +d))^(1/2)*EllipticF((2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)*(g*x+f)^(1/2)/ (2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(e*x+d)^(1/2),((2*c*d-(b+(-4*a*c+b^ 2)^(1/2))*e)*(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)/(2*c*d-(b-(-4*a*c+b^2)^(1/2) )*e)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g))^(1/2))/e^2/(2*c*d-(b-(-4*a*c+b^2)^( 1/2))*e)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b-(-4*a*c+b^2)^(1/ 2))*g)^(1/2)/((-d*g+e*f)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f-(b-(-4*a*c+b^ 2)^(1/2))*g)/(e*x+d))^(1/2)/(c*x^2+b*x+a)^(1/2)+2*C*(-d*g+e*f)*(b-(-4*a*c+ b^2)^(1/2)+2*c*x)*((-d*g+e*f)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f-(b+(-4*a *c+b^2)^(1/2))*g)/(e*x+d))^(1/2)*EllipticPi((2*c*d-(b-(-4*a*c+b^2)^(1/2...
Leaf count is larger than twice the leaf count of optimal. \(4412\) vs. \(2(1384)=2768\).
Time = 36.36 (sec) , antiderivative size = 4412, normalized size of antiderivative = 3.19 \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(A + C*x^2)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2] ),x]
Output:
(-2*(C*d^2 + A*e^2)*Sqrt[f + g*x]*(a + b*x + c*x^2))/((c*d^2 - b*d*e + a*e ^2)*(e*f - d*g)*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]) + (2*Sqrt[a + b*x + c *x^2]*(((-(C*d^2) - A*e^2)*(d + e*x)^(5/2)*(c + (c*d^2)/(d + e*x)^2 - (b*d *e)/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x ))*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x)))/(Sqrt[((d + e*x)^2*(c*(-1 + d/ (d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2 ]*Sqrt[f + ((d + e*x)*(g - (d*g)/(d + e*x)))/e]) - ((c*d^2 - b*d*e + a*e^2 )*(-(e*f) + d*g)*(d + e*x)^(3/2)*Sqrt[(c + (c*d^2)/(d + e*x)^2 - (b*d*e)/( d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x))*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x))]*((-2*C*d*(-1/2*(2*c*d - b*e - Sqrt[ b^2*e^2 - 4*a*c*e^2])/(c*d^2 - b*d*e + a*e^2) + (d + e*x)^(-1))*Sqrt[(-1/2 *(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])/(c*d^2 - b*d*e + a*e^2) + (d + e*x)^(-1))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a*e^2)) - (2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a* e^2)))]*Sqrt[(-(g/(-(e*f) + d*g)) + (d + e*x)^(-1))/((2*c*d - b*e - Sqrt[b ^2*e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a*e^2)) - g/(-(e*f) + d*g))]*Elli pticF[ArcSin[Sqrt[(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*d^2)/(d + e*x) - (2*b*d*e)/(d + e*x) + (2*a*e^2)/(d + e*x))/Sqrt[(b^2 - 4*a*c)*e^2] ]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*e^2]*(e*f - d*g))/(-2*c*d*e*f + b*e^2*f + e*Sqrt[(b^2 - 4*a*c)*e^2]*f + b*d*e*g - 2*a*e^2*g - d*Sqrt[(b^2 - 4*a...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 2154 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx\) |
\(\Big \downarrow \) 1281 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {g \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx\) |
\(\Big \downarrow \) 1280 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {2 g (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a+b x+c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx\) |
\(\Big \downarrow \) 1292 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {2 g (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a+b x+c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt {a+b x+c x^2} (e f-d g)^2 \sqrt [4]{a e^2-b d e+c d^2} \sqrt {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx\) |
\(\Big \downarrow \) 2154 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt {a+b x+c x^2} (e f-d g)^2 \sqrt [4]{a e^2-b d e+c d^2} \sqrt {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}}\right )-\frac {2 C d \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}+\int \frac {C \sqrt {d+e x}}{e^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt {a+b x+c x^2} (e f-d g)^2 \sqrt [4]{a e^2-b d e+c d^2} \sqrt {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}}\right )-\frac {2 C d \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}+\frac {C \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}\) |
\(\Big \downarrow \) 1276 |
\(\displaystyle \frac {\sqrt {2} C \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {\frac {(e f-d g) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g},\arcsin \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{e^2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {x c+\frac {2 a c}{b+\sqrt {b^2-4 a c}}} \sqrt {c x^2+b x+a}}-\frac {2 C d \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e^2}+\left (\frac {C d^2}{e^2}+A\right ) \left (\frac {g \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt [4]{c d^2-b e d+a e^2} (e f-d g)^2 \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )\) |
\(\Big \downarrow \) 1280 |
\(\displaystyle \frac {\sqrt {2} C \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {\frac {(e f-d g) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g},\arcsin \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{e^2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {x c+\frac {2 a c}{b+\sqrt {b^2-4 a c}}} \sqrt {c x^2+b x+a}}+\left (\frac {C d^2}{e^2}+A\right ) \left (\frac {g \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt [4]{c d^2-b e d+a e^2} (e f-d g)^2 \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )+\frac {4 C d (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{e^2 (e f-d g) \sqrt {c x^2+b x+a}}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {2 C d \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{e^2 \sqrt [4]{c d^2-b e d+a e^2} (e f-d g) \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}+\frac {\sqrt {2} C \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {\frac {(e f-d g) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g},\arcsin \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{e^2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {x c+\frac {2 a c}{b+\sqrt {b^2-4 a c}}} \sqrt {c x^2+b x+a}}+\left (\frac {C d^2}{e^2}+A\right ) \left (\frac {g \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt [4]{c d^2-b e d+a e^2} (e f-d g)^2 \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )\) |
Input:
Int[(A + C*x^2)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(d_.) + (e_.)*(x_)]/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*( x_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt [2]*Sqrt[2*c*f - g*(b + q)]*Sqrt[b - q + 2*c*x]*(d + e*x)*Sqrt[(e*f - d*g)* ((b + q + 2*c*x)/((2*c*f - g*(b + q))*(d + e*x)))]*(Sqrt[(e*f - d*g)*((2*a + (b + q)*x)/((b*f + q*f - 2*a*g)*(d + e*x)))]/(g*Sqrt[2*c*d - e*(b + q)]*S qrt[2*a*(c/(b + q)) + c*x]*Sqrt[a + b*x + c*x^2]))*EllipticPi[e*((2*c*f - g *(b + q))/(g*(2*c*d - e*(b + q)))), ArcSin[Sqrt[2*c*d - e*(b + q)]*(Sqrt[f + g*x]/(Sqrt[2*c*f - g*(b + q)]*Sqrt[d + e*x]))], (b*d + q*d - 2*a*e)*((2*c *f - g*(b + q))/((b*f + q*f - 2*a*g)*(2*c*d - e*(b + q))))], x]] /; FreeQ[{ a, b, c, d, e, f, g}, x]
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.) *(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-2*(d + e*x)*(Sqrt[(e*f - d*g)^2* ((a + b*x + c*x^2)/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2))]/((e*f - d*g)*Sqr t[a + b*x + c*x^2])) Subst[Int[1/Sqrt[1 - (2*c*d*f - b*e*f - b*d*g + 2*a* e*g)*(x^2/(c*f^2 - b*f*g + a*g^2)) + (c*d^2 - b*d*e + a*e^2)*(x^4/(c*f^2 - b*f*g + a*g^2))], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c , d, e, f, g}, x]
Int[1/(((d_.) + (e_.)*(x_))^(3/2)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_ .)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-g/(e*f - d*g) Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] + Simp[e/(e*f - d*g) Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[ {a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* (a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(3176\) vs. \(2(1243)=2486\).
Time = 15.03 (sec) , antiderivative size = 3177, normalized size of antiderivative = 2.30
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3177\) |
default | \(\text {Expression too large to display}\) | \(140616\) |
Input:
int((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RE TURNVERBOSE)
Output:
((g*x+f)*(c*x^2+b*x+a)*(e*x+d))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)/(e *x+d)^(1/2)*(2*(c*e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f)/(a*d* e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)/e*(A*e^2+C*d^2)/((x+d /e)*(c*e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f))^(1/2)+2*(-C*d/e ^2+1/e^2*(a*e^2*g-b*d*e*g+b*e^2*f+c*d^2*g-c*d*e*f)*(A*e^2+C*d^2)/(a*d*e^2* g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)-(a*e*g+b*e*f)/(a*d*e^2*g- a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)/e*(A*e^2+C*d^2))*(-f/g+d/e) *((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+f/g)/(f/g-d/e)/(x-1/2/c*(-b+(-4* a*c+b^2)^(1/2))))^(1/2)*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*((1/2/c*(-b+(- 4*a*c+b^2)^(1/2))+f/g)*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-1/2*(b+(-4*a*c+b ^2)^(1/2))/c+f/g)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((1/2/c*(-b+(-4 *a*c+b^2)^(1/2))+f/g)*(x+d/e)/(f/g-d/e)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))) ^(1/2)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(1/2/c*(-b+(-4*a*c+b^2)^(1/2)) +f/g)/(c*e*g*(x+f/g)*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+1/2*(b+(-4*a*c+b ^2)^(1/2))/c)*(x+d/e))^(1/2)*EllipticF(((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2) ))*(x+f/g)/(f/g-d/e)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2),((1/2/c*(-b+ (-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*(-f/g+d/e)/(1/2*(b+(-4*a *c+b^2)^(1/2))/c-f/g)/(d/e+1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*(C/e+1 /e*(b*e*g-c*d*g+c*e*f)*(A*e^2+C*d^2)/(a*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2* f+c*d^3*g-c*d^2*e*f)-(2*b*e*g+2*c*e*f)/(a*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d...
Timed out. \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, alg orithm="fricas")
Output:
Timed out
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + C x^{2}}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((C*x**2+A)/(e*x+d)**(3/2)/(g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x )
Output:
Integral((A + C*x**2)/((d + e*x)**(3/2)*sqrt(f + g*x)*sqrt(a + b*x + c*x** 2)), x)
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:
integrate((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, alg orithm="maxima")
Output:
integrate((C*x^2 + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)*sqrt(g*x + f) ), x)
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:
integrate((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, alg orithm="giac")
Output:
integrate((C*x^2 + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)*sqrt(g*x + f) ), x)
Timed out. \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {C\,x^2+A}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((A + C*x^2)/((f + g*x)^(1/2)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)), x)
Output:
int((A + C*x^2)/((f + g*x)^(1/2)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {C \,x^{2}+A}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}d x \] Input:
int((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
int((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x)