\(\int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx\) [49]

Optimal result

Integrand size = 40, antiderivative size = 1060 \[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx =\text {Too large to display} \] Output:

2*d*(-B*e+C*d)*(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*(c^(1/2)*C*d^2-a^(1/2)*e*(-B*e+2*C*d))*(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))^...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3448\) vs. \(2(1060)=2120\).

Time = 34.06 (sec) , antiderivative size = 3448, normalized size of antiderivative = 3.25 \[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(B*x + C*x^2)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a - c*x^2]),x]
 

Output:

(-2*d*(C*d - B*e)*Sqrt[f + g*x]*Sqrt[a - c*x^2])/((-(c*d^2) + a*e^2)*(e*f 
- d*g)*Sqrt[d + e*x]) - (2*(-((d*(C*d - B*e)*Sqrt[d + e*x]*(-c - (c*d^2)/( 
d + e*x)^2 + (a*e^2)/(d + e*x)^2 + (2*c*d)/(d + e*x))*(g + (e*f)/(d + e*x) 
 - (d*g)/(d + e*x))*Sqrt[a - (c*(d + e*x)^2*(-1 + d/(d + e*x))^2)/e^2])/(( 
(a*e^2)/(d + e*x)^2 - c*(-1 + d/(d + e*x))^2)*Sqrt[f + ((d + e*x)*(g - (d* 
g)/(d + e*x)))/e])) + ((c*d^2 - a*e^2)*(-(e*f) + d*g)*Sqrt[(-c - (c*d^2)/( 
d + e*x)^2 + (a*e^2)/(d + e*x)^2 + (2*c*d)/(d + e*x))*(g + (e*f)/(d + e*x) 
 - (d*g)/(d + e*x))]*Sqrt[a - (c*(d + e*x)^2*(-1 + d/(d + e*x))^2)/e^2]*(( 
-2*C*d*Sqrt[(-((-(c*d) - Sqrt[a]*Sqrt[c]*e)/(-(c*d^2) + a*e^2)) + (d + e*x 
)^(-1))/(-((-(c*d) - Sqrt[a]*Sqrt[c]*e)/(-(c*d^2) + a*e^2)) + (-(c*d) + Sq 
rt[a]*Sqrt[c]*e)/(-(c*d^2) + a*e^2))]*(-((-(c*d) + Sqrt[a]*Sqrt[c]*e)/(-(c 
*d^2) + a*e^2)) + (d + e*x)^(-1))*Sqrt[(-(g/(-(e*f) + d*g)) + (d + e*x)^(- 
1))/((-(c*d) + Sqrt[a]*Sqrt[c]*e)/(-(c*d^2) + a*e^2) - g/(-(e*f) + d*g))]* 
EllipticF[ArcSin[Sqrt[((-(e*f) + d*g)*(-Sqrt[c] + (Sqrt[c]*d)/(d + e*x) + 
(Sqrt[a]*e)/(d + e*x)))/(e*(Sqrt[c]*f + Sqrt[a]*g))]], ((Sqrt[c]*d - Sqrt[ 
a]*e)*(Sqrt[c]*f + Sqrt[a]*g))/(2*Sqrt[a]*Sqrt[c]*(-(e*f) + d*g))])/(Sqrt[ 
(-((-(c*d) + Sqrt[a]*Sqrt[c]*e)/(-(c*d^2) + a*e^2)) + (d + e*x)^(-1))/(-(( 
-(c*d) + Sqrt[a]*Sqrt[c]*e)/(-(c*d^2) + a*e^2)) + g/(-(e*f) + d*g))]*Sqrt[ 
(c + (c*d^2 - a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x))*(-g + (-(e*f) + d*g) 
/(d + e*x))]) + (B*e*Sqrt[(-((-(c*d) - Sqrt[a]*Sqrt[c]*e)/(-(c*d^2) + a...
 

Rubi [F]

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.

\(\displaystyle \int \frac {B x+C x^2}{\sqrt {a-c x^2} (d+e x)^{3/2} \sqrt {f+g x}} \, dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \frac {x (B+C x)}{\sqrt {a-c x^2} (d+e x)^{3/2} \sqrt {f+g x}}dx\)

\(\Big \downarrow \) 2349

\(\displaystyle \frac {d (C d-B e) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}+\int \frac {\frac {B}{e}+\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\)

\(\Big \downarrow \) 733

\(\displaystyle \int \frac {\frac {B}{e}+\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx+\frac {d (C d-B e) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}-\frac {g \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e f-d g}\right )}{e^2}\)

\(\Big \downarrow \) 732

\(\displaystyle \frac {d (C d-B e) \left (\frac {2 g (d+e x) \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a-c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )}{e^2}+\int \frac {\frac {B}{e}+\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\)

\(\Big \downarrow \) 744

\(\displaystyle \frac {d (C d-B e) \left (\frac {2 g (d+e x) \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a-c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )}{e^2}+\int \frac {\frac {B}{e}+\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {d (C d-B e) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-a g^2} \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}{\left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt {a-c x^2} \sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}}\right )}{e^2}+\int \frac {\frac {B}{e}+\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx\)

\(\Big \downarrow \) 2349

\(\displaystyle \frac {d (C d-B e) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-a g^2} \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}{\left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt {a-c x^2} \sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}}\right )}{e^2}-\frac {(2 C d-B e) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}+\int \frac {C \sqrt {d+e x}}{e^2 \sqrt {f+g x} \sqrt {a-c x^2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d (C d-B e) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-a g^2} \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}{\left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt {a-c x^2} \sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}}\right )}{e^2}-\frac {(2 C d-B e) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}+\frac {C \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}\)

\(\Big \downarrow \) 726

\(\displaystyle \frac {2 C \sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (\sqrt {c} x+\sqrt {a}\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) g},\arcsin \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {f+g x}}{\sqrt {-c f-\sqrt {a} \sqrt {c} g} \sqrt {d+e x}}\right ),\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{e^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e} g \sqrt {a-c x^2}}-\frac {(2 C d-B e) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e^2}+\frac {d (C d-B e) \left (\frac {g \sqrt [4]{c f^2-a g^2} (d+e x) \sqrt {-\frac {(e f-d g)^2 \left (a-c x^2\right )}{\left (c f^2-a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-a e^2} (f+g x)}{\sqrt {c f^2-a g^2} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {a-c x^2} \sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\right )}{e^2}\)

\(\Big \downarrow \) 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}}+\frac {d (C d-B e) \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 )}{e^2}+\frac {2 (2 C d-B e) (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-B e) \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}}+\frac {d (C d-B e) \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 )}{e^2}\)

Input:

Int[(B*x + C*x^2)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a - c*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 726

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]
 

rule 732

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]
 

rule 733

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]
 

rule 744

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]
 

rule 1416

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]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2349

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2207\) vs. \(2(825)=1650\).

Time = 7.53 (sec) , antiderivative size = 2208, normalized size of antiderivative = 2.08

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2208\)
default	\(\text {Expression too large to display}\)	\(29044\)

Input:

int((C*x^2+B*x)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x,method=_RET 
URNVERBOSE)
 

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*d*(B*e-C*d)/((x+d/e)*(-c*e*g*x^3-c*e*f*x^2+a*e*g*x+a*e*f))^(1 
/2)+2*((B*e-C*d)/e^2-d*(B*e-C*d)*(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)*d*(B*e-C*d))*(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*d*(B*e-C*d)/(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)*d*(B*e-C*d))*(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)^(1...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\text {Timed out} \] Input:

integrate((C*x^2+B*x)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algo 
rithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {x \left (B + C x\right )}{\sqrt {a - c x^{2}} \left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \] Input:

integrate((C*x**2+B*x)/(e*x+d)**(3/2)/(g*x+f)**(1/2)/(-c*x**2+a)**(1/2),x)
 

Output:

Integral(x*(B + C*x)/(sqrt(a - c*x**2)*(d + e*x)**(3/2)*sqrt(f + g*x)), x)
 

Maxima [F]

\[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int { \frac {C x^{2} + B x}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:

integrate((C*x^2+B*x)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algo 
rithm="maxima")
 

Output:

integrate((C*x^2 + B*x)/(sqrt(-c*x^2 + a)*(e*x + d)^(3/2)*sqrt(g*x + f)), 
x)
 

Giac [F]

\[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int { \frac {C x^{2} + B x}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:

integrate((C*x^2+B*x)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algo 
rithm="giac")
 

Output:

integrate((C*x^2 + B*x)/(sqrt(-c*x^2 + a)*(e*x + d)^(3/2)*sqrt(g*x + f)), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {C\,x^2+B\,x}{\sqrt {f+g\,x}\,\sqrt {a-c\,x^2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:

int((B*x + C*x^2)/((f + g*x)^(1/2)*(a - c*x^2)^(1/2)*(d + e*x)^(3/2)),x)
 

Output:

int((B*x + C*x^2)/((f + g*x)^(1/2)*(a - c*x^2)^(1/2)*(d + e*x)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {B x+C x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {C \,x^{2}+B x}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {g x +f}\, \sqrt {-c \,x^{2}+a}}d x \] Input:

int((C*x^2+B*x)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x)
 

Output:

int((C*x^2+B*x)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x)