Integrand size = 38, antiderivative size = 1064 \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx =\text {Too large to display} \] Output:
2*(A*e^2+C*d^2)*(c^(1/2)*f+a^(1/2)*g)^(1/2)*(a^(1/2)+c^(1/2)*x)*(-(-d*g+e* f)*(a^(1/2)-c^(1/2)*x)/(c^(1/2)*f+a^(1/2)*g)/(e*x+d))^(1/2)*EllipticE((c^( 1/2)*d+a^(1/2)*e)^(1/2)*(g*x+f)^(1/2)/(c^(1/2)*f+a^(1/2)*g)^(1/2)/(e*x+d)^ (1/2),((c^(1/2)*d-a^(1/2)*e)*(c^(1/2)*f+a^(1/2)*g)/(c^(1/2)*d+a^(1/2)*e)/( c^(1/2)*f-a^(1/2)*g))^(1/2))/e/(c^(1/2)*d-a^(1/2)*e)/(c^(1/2)*d+a^(1/2)*e) ^(1/2)/(-d*g+e*f)/((-d*g+e*f)*(a^(1/2)+c^(1/2)*x)/(c^(1/2)*f-a^(1/2)*g)/(e *x+d))^(1/2)/(-c*x^2+a)^(1/2)-2*(2*a^(1/2)*C*d*e-c^(1/2)*(-A*e^2+C*d^2))*( c^(1/2)*f+a^(1/2)*g)^(1/2)*(a^(1/2)+c^(1/2)*x)*(-(-d*g+e*f)*(a^(1/2)-c^(1/ 2)*x)/(c^(1/2)*f+a^(1/2)*g)/(e*x+d))^(1/2)*EllipticF((c^(1/2)*d+a^(1/2)*e) ^(1/2)*(g*x+f)^(1/2)/(c^(1/2)*f+a^(1/2)*g)^(1/2)/(e*x+d)^(1/2),((c^(1/2)*d -a^(1/2)*e)*(c^(1/2)*f+a^(1/2)*g)/(c^(1/2)*d+a^(1/2)*e)/(c^(1/2)*f-a^(1/2) *g))^(1/2))/e^2/(c^(1/2)*d-a^(1/2)*e)/(c^(1/2)*d+a^(1/2)*e)^(1/2)/(c^(1/2) *f-a^(1/2)*g)/((-d*g+e*f)*(a^(1/2)+c^(1/2)*x)/(c^(1/2)*f-a^(1/2)*g)/(e*x+d ))^(1/2)/(-c*x^2+a)^(1/2)+2*C*(c^(1/2)*f+a^(1/2)*g)^(1/2)*(-d*g+e*f)*(a^(1 /2)+c^(1/2)*x)*(-(-d*g+e*f)*(a^(1/2)-c^(1/2)*x)/(c^(1/2)*f+a^(1/2)*g)/(e*x +d))^(1/2)*EllipticPi((c^(1/2)*d+a^(1/2)*e)^(1/2)*(g*x+f)^(1/2)/(c^(1/2)*f +a^(1/2)*g)^(1/2)/(e*x+d)^(1/2),e*(c^(1/2)*f+a^(1/2)*g)/(c^(1/2)*d+a^(1/2) *e)/g,((c^(1/2)*d-a^(1/2)*e)*(c^(1/2)*f+a^(1/2)*g)/(c^(1/2)*d+a^(1/2)*e)/( c^(1/2)*f-a^(1/2)*g))^(1/2))/e^2/(c^(1/2)*d+a^(1/2)*e)^(1/2)/g/(c^(1/2)*f- a^(1/2)*g)/((-d*g+e*f)*(a^(1/2)+c^(1/2)*x)/(c^(1/2)*f-a^(1/2)*g)/(e*x+d...
Leaf count is larger than twice the leaf count of optimal. \(2191\) vs. \(2(1064)=2128\).
Time = 33.62 (sec) , antiderivative size = 2191, normalized size of antiderivative = 2.06 \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(A + C*x^2)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a - c*x^2]),x]
Output:
(-2*(C*d^2 + A*e^2)*Sqrt[f + g*x]*Sqrt[a - c*x^2])/((-(c*d^2) + a*e^2)*(e* f - d*g)*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*Sqrt[a - (c*(d + e*x)^2*(-1 + d /(d + e*x))^2)/e^2]*(-((C*d^2 + A*e^2)*(g + (e*f)/(d + e*x) - (d*g)/(d + e *x))*(-((a*e^2)/(d + e*x)^2) + c*(-1 + d/(d + e*x))^2)) + (-2*C*d^2*(c*d - Sqrt[a]*Sqrt[c]*e)*Sqrt[((Sqrt[c]*d + Sqrt[a]*e)*(g + (e*f)/(d + e*x) - ( d*g)/(d + e*x)))/(e*(Sqrt[c]*f + Sqrt[a]*g))]*Sqrt[((e*f - d*g)*((Sqrt[a]* e)/(d + e*x) + Sqrt[c]*(1 - d/(d + e*x))))/(e*(Sqrt[c]*f - Sqrt[a]*g))]*(- (Sqrt[a]*Sqrt[c]*e) + (a*e^2)/(d + e*x) + c*(d - d^2/(d + e*x)))*(e*(c*d^2 - a*e^2)*(Sqrt[c]*f - Sqrt[a]*g)*EllipticE[ArcSin[Sqrt[((Sqrt[c]*d + Sqrt [a]*e)*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x)))/(e*(Sqrt[c]*f + Sqrt[a]*g) )]], ((Sqrt[c]*d - Sqrt[a]*e)*(Sqrt[c]*f + Sqrt[a]*g))/((Sqrt[c]*d + Sqrt[ a]*e)*(Sqrt[c]*f - Sqrt[a]*g))] + (Sqrt[c]*d - Sqrt[a]*e)*(c*d + Sqrt[a]*S qrt[c]*e)*(-(e*f) + d*g)*EllipticF[ArcSin[Sqrt[((Sqrt[c]*d + Sqrt[a]*e)*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x)))/(e*(Sqrt[c]*f + Sqrt[a]*g))]], ((Sq rt[c]*d - Sqrt[a]*e)*(Sqrt[c]*f + Sqrt[a]*g))/((Sqrt[c]*d + Sqrt[a]*e)*(Sq rt[c]*f - Sqrt[a]*g))]) - 2*A*e^2*(c*d - Sqrt[a]*Sqrt[c]*e)*Sqrt[((Sqrt[c] *d + Sqrt[a]*e)*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x)))/(e*(Sqrt[c]*f + S qrt[a]*g))]*Sqrt[((e*f - d*g)*((Sqrt[a]*e)/(d + e*x) + Sqrt[c]*(1 - d/(d + e*x))))/(e*(Sqrt[c]*f - Sqrt[a]*g))]*(-(Sqrt[a]*Sqrt[c]*e) + (a*e^2)/(d + e*x) + c*(d - d^2/(d + e*x)))*(e*(c*d^2 - a*e^2)*(Sqrt[c]*f - Sqrt[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.
\(\displaystyle \int \frac {A+C x^2}{\sqrt {a-c x^2} (d+e x)^{3/2} \sqrt {f+g x}} \, dx\) |
\(\Big \downarrow \) 2349 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}}dx+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\) |
\(\Big \downarrow \) 733 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}-\frac {g \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e f-d g}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\) |
\(\Big \downarrow \) 732 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {2 g (d+e x) \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a-c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\) |
\(\Big \downarrow \) 744 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {2 g (d+e x) \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a-c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-a g^2} \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}{\left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt {a-c x^2} \sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}}\right )+\int \frac {\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\) |
\(\Big \downarrow \) 2349 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-a g^2} \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}{\left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt {a-c x^2} \sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}}\right )-\frac {2 C d \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}+\int \frac {C \sqrt {d+e x}}{e^2 \sqrt {f+g x} \sqrt {a-c x^2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (A+\frac {C d^2}{e^2}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-a g^2} \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}{\left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt {a-c x^2} \sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}}\right )-\frac {2 C d \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}+\frac {C \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}\) |
\(\Big \downarrow \) 726 |
\(\displaystyle \frac {2 C \sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (\sqrt {c} x+\sqrt {a}\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) g},\arcsin \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {f+g x}}{\sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {d+e x}}\right ),\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{e^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e} g \sqrt {a-c x^2}}-\frac {2 C d \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}+\left (\frac {C d^2}{e^2}+A\right ) \left (\frac {g \sqrt [4]{c f^2-a g^2} (d+e x) \sqrt {-\frac {(e f-d g)^2 \left (a-c x^2\right )}{\left (c f^2-a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {a-c x^2} \sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )\) |
\(\Big \downarrow \) 732 |
\(\displaystyle \frac {2 C \sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (\sqrt {c} x+\sqrt {a}\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) g},\arcsin \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {f+g x}}{\sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {d+e x}}\right ),\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{e^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e} g \sqrt {a-c x^2}}+\left (\frac {C d^2}{e^2}+A\right ) \left (\frac {g \sqrt [4]{c f^2-a g^2} (d+e x) \sqrt {-\frac {(e f-d g)^2 \left (a-c x^2\right )}{\left (c f^2-a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {a-c x^2} \sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )+\frac {4 C d (d+e x) \sqrt {-\frac {(e f-d g)^2 \left (a-c x^2\right )}{\left (c f^2-a g^2\right ) (d+e x)^2}} \int \frac {1}{\sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-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 {a-c x^2}}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {2 C d \sqrt [4]{c f^2-a g^2} (d+e x) \sqrt {-\frac {(e f-d g)^2 \left (a-c x^2\right )}{\left (c f^2-a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{e^2 \sqrt [4]{c d^2-a e^2} (e f-d g) \sqrt {a-c x^2} \sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}+\frac {2 C \sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (\sqrt {c} x+\sqrt {a}\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) g},\arcsin \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {f+g x}}{\sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {d+e x}}\right ),\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{e^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e} g \sqrt {a-c x^2}}+\left (\frac {C d^2}{e^2}+A\right ) \left (\frac {g \sqrt [4]{c f^2-a g^2} (d+e x) \sqrt {-\frac {(e f-d g)^2 \left (a-c x^2\right )}{\left (c f^2-a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {a-c x^2} \sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )\) |
Input:
Int[(A + C*x^2)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a - 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_) + (c_.)*(x _)^2]), x_Symbol] :> With[{q = Rt[-4*a*c, 2]}, Simp[Sqrt[2]*Sqrt[2*c*f - g* q]*Sqrt[-q + 2*c*x]*(d + e*x)*Sqrt[(e*f - d*g)*((q + 2*c*x)/((2*c*f - g*q)* (d + e*x)))]*(Sqrt[(e*f - d*g)*((2*a + q*x)/((q*f - 2*a*g)*(d + e*x)))]/(g* Sqrt[2*c*d - e*q]*Sqrt[2*a*(c/q) + c*x]*Sqrt[a + c*x^2]))*EllipticPi[e*((2* c*f - g*q)/(g*(2*c*d - e*q))), ArcSin[Sqrt[2*c*d - e*q]*(Sqrt[f + g*x]/(Sqr t[2*c*f - g*q]*Sqrt[d + e*x]))], (q*d - 2*a*e)*((2*c*f - g*q)/((q*f - 2*a*g )*(2*c*d - e*q)))], x]] /; FreeQ[{a, c, d, e, f, g}, x]
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)* (x_)^2]), x_Symbol] :> Simp[-2*(c + d*x)*(Sqrt[(d*e - c*f)^2*((a + b*x^2)/( (b*e^2 + a*f^2)*(c + d*x)^2))]/((d*e - c*f)*Sqrt[a + b*x^2])) Subst[Int[1 /Sqrt[Simp[1 - (2*b*c*e + 2*a*d*f)*(x^2/(b*e^2 + a*f^2)) + (b*c^2 + a*d^2)* (x^4/(b*e^2 + a*f^2)), x]], x], x, Sqrt[e + f*x]/Sqrt[c + d*x]], x] /; Free Q[{a, b, c, d, e, f}, x]
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/2)*Sqrt[(a_) + (b_. )*(x_)^2]), x_Symbol] :> Simp[d/(d*e - c*f) Int[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[a + b*x^2]), x], x] - Simp[f/(d*e - c*f) Int[Sqrt[c + d*x]/((e + f*x)^(3/2)*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x] /; FreeQ[{a, 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_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d *x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, 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. \(2216\) vs. \(2(829)=1658\).
Time = 7.51 (sec) , antiderivative size = 2217, normalized size of antiderivative = 2.08
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(2217\) |
default | \(\text {Expression too large to display}\) | \(28952\) |
Input:
int((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x,method=_RETUR NVERBOSE)
Output:
((e*x+d)*(g*x+f)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(1/2)/(-c*x^2+a)^ (1/2)*(2*(-c*e*g*x^3-c*e*f*x^2+a*e*g*x+a*e*f)/(a*d*e^2*g-a*e^3*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-c*e*f*x^2+a*e*g*x+a*e*f))^( 1/2)+2*(-C*d/e^2+(A*e^2+C*d^2)*(a*e^2*g-c*d^2*g+c*d*e*f)/e^2/(a*e^2-c*d^2) /(d*g-e*f)-a*g/(a*d*e^2*g-a*e^3*f-c*d^3*g+c*d^2*e*f)*(A*e^2+C*d^2))*(1/c*( a*c)^(1/2)-d/e)*((-1/c*(a*c)^(1/2)+f/g)*(x+d/e)/(-1/c*(a*c)^(1/2)+d/e)/(x+ f/g))^(1/2)*(x+f/g)^2*((-f/g+d/e)*(x-1/c*(a*c)^(1/2))/(1/c*(a*c)^(1/2)+d/e )/(x+f/g))^(1/2)*((-f/g+d/e)*(x+1/c*(a*c)^(1/2))/(-1/c*(a*c)^(1/2)+d/e)/(x +f/g))^(1/2)/(-1/c*(a*c)^(1/2)+f/g)/(-f/g+d/e)/(-c*e*g*(x+d/e)*(x+f/g)*(x- 1/c*(a*c)^(1/2))*(x+1/c*(a*c)^(1/2)))^(1/2)*EllipticF(((-1/c*(a*c)^(1/2)+f /g)*(x+d/e)/(-1/c*(a*c)^(1/2)+d/e)/(x+f/g))^(1/2),((-f/g-1/c*(a*c)^(1/2))* (1/c*(a*c)^(1/2)-d/e)/(-1/c*(a*c)^(1/2)-d/e)/(1/c*(a*c)^(1/2)-f/g))^(1/2)) +2*(C/e+1/e*c*(A*e^2+C*d^2)/(a*e^2-c*d^2)+2*c*f/(a*d*e^2*g-a*e^3*f-c*d^3*g +c*d^2*e*f)*(A*e^2+C*d^2))*(1/c*(a*c)^(1/2)-d/e)*((-1/c*(a*c)^(1/2)+f/g)*( x+d/e)/(-1/c*(a*c)^(1/2)+d/e)/(x+f/g))^(1/2)*(x+f/g)^2*((-f/g+d/e)*(x-1/c* (a*c)^(1/2))/(1/c*(a*c)^(1/2)+d/e)/(x+f/g))^(1/2)*((-f/g+d/e)*(x+1/c*(a*c) ^(1/2))/(-1/c*(a*c)^(1/2)+d/e)/(x+f/g))^(1/2)/(-1/c*(a*c)^(1/2)+f/g)/(-f/g +d/e)/(-c*e*g*(x+d/e)*(x+f/g)*(x-1/c*(a*c)^(1/2))*(x+1/c*(a*c)^(1/2)))^(1/ 2)*(-f/g*EllipticF(((-1/c*(a*c)^(1/2)+f/g)*(x+d/e)/(-1/c*(a*c)^(1/2)+d/e)/ (x+f/g))^(1/2),((-f/g-1/c*(a*c)^(1/2))*(1/c*(a*c)^(1/2)-d/e)/(-1/c*(a*c...
Timed out. \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algori thm="fricas")
Output:
Timed out
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {A + C x^{2}}{\sqrt {a - c x^{2}} \left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \] Input:
integrate((C*x**2+A)/(e*x+d)**(3/2)/(g*x+f)**(1/2)/(-c*x**2+a)**(1/2),x)
Output:
Integral((A + C*x**2)/(sqrt(a - c*x**2)*(d + e*x)**(3/2)*sqrt(f + g*x)), x )
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:
integrate((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algori thm="maxima")
Output:
integrate((C*x^2 + A)/(sqrt(-c*x^2 + a)*(e*x + d)^(3/2)*sqrt(g*x + f)), x)
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:
integrate((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algori thm="giac")
Output:
integrate((C*x^2 + A)/(sqrt(-c*x^2 + a)*(e*x + d)^(3/2)*sqrt(g*x + f)), x)
Timed out. \[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {C\,x^2+A}{\sqrt {f+g\,x}\,\sqrt {a-c\,x^2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:
int((A + C*x^2)/((f + g*x)^(1/2)*(a - c*x^2)^(1/2)*(d + e*x)^(3/2)),x)
Output:
int((A + C*x^2)/((f + g*x)^(1/2)*(a - c*x^2)^(1/2)*(d + e*x)^(3/2)), x)
\[ \int \frac {A+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {C \,x^{2}+A}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {g x +f}\, \sqrt {-c \,x^{2}+a}}d x \] Input:
int((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x)
Output:
int((C*x^2+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x)