Integrand size = 28, antiderivative size = 1038 \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\frac {e^2 \sqrt {c+d x} \sqrt {a+b x^2}}{b d \sqrt {e x}}-\frac {\sqrt {-a} \left (\sqrt {b} c+\sqrt {-a} d\right ) \sqrt {\sqrt {-a} \sqrt {b} c+a d} e^{3/2} x \sqrt {\frac {c^2 \left (a+b x^2\right )}{\left (b c^2+a d^2\right ) x^2}} \sqrt {1+\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c-a d\right ) x}} \sqrt {1-\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c+a d\right ) x}} E\left (\arcsin \left (\frac {\sqrt {a} \sqrt {e} \sqrt {c+d x}}{\sqrt {\sqrt {-a} \sqrt {b} c+a d} \sqrt {e x}}\right )|-\frac {\sqrt {-a} \sqrt {b} c+a d}{\sqrt {-a} \sqrt {b} c-a d}\right )}{\sqrt {a} b c d \sqrt {a+b x^2} \sqrt {1-\frac {2 a d (c+d x)}{\left (b c^2+a d^2\right ) x}+\frac {a (c+d x)^2}{\left (b c^2+a d^2\right ) x^2}}}-\frac {\sqrt {a} \left (\sqrt {b} c+\sqrt {-a} d\right ) \sqrt {\sqrt {-a} \sqrt {b} c+a d} e^{3/2} x \sqrt {\frac {c^2 \left (a+b x^2\right )}{\left (b c^2+a d^2\right ) x^2}} \sqrt {1+\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c-a d\right ) x}} \sqrt {1-\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c+a d\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {e} \sqrt {c+d x}}{\sqrt {\sqrt {-a} \sqrt {b} c+a d} \sqrt {e x}}\right ),-\frac {\sqrt {-a} \sqrt {b} c+a d}{\sqrt {-a} \sqrt {b} c-a d}\right )}{\sqrt {b} d \left (\sqrt {-a} \sqrt {b} c-a d\right ) \sqrt {a+b x^2} \sqrt {1-\frac {2 a d (c+d x)}{\left (b c^2+a d^2\right ) x}+\frac {a (c+d x)^2}{\left (b c^2+a d^2\right ) x^2}}}-\frac {c \sqrt {\sqrt {-a} \sqrt {b} c+a d} e^{3/2} x \sqrt {\frac {c^2 \left (a+b x^2\right )}{\left (b c^2+a d^2\right ) x^2}} \sqrt {1+\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c-a d\right ) x}} \sqrt {1-\frac {a (c+d x)}{\left (\sqrt {-a} \sqrt {b} c+a d\right ) x}} \operatorname {EllipticPi}\left (1+\frac {a \sqrt {b} c}{(-a)^{3/2} d},\arcsin \left (\frac {\sqrt {a} \sqrt {e} \sqrt {c+d x}}{\sqrt {\sqrt {-a} \sqrt {b} c+a d} \sqrt {e x}}\right ),-\frac {\sqrt {-a} \sqrt {b} c+a d}{\sqrt {-a} \sqrt {b} c-a d}\right )}{\sqrt {a} d^2 \sqrt {a+b x^2} \sqrt {1-\frac {2 a d (c+d x)}{\left (b c^2+a d^2\right ) x}+\frac {a (c+d x)^2}{\left (b c^2+a d^2\right ) x^2}}} \] Output:
e^2*(d*x+c)^(1/2)*(b*x^2+a)^(1/2)/b/d/(e*x)^(1/2)-(-a)^(1/2)*(b^(1/2)*c+(- a)^(1/2)*d)*((-a)^(1/2)*b^(1/2)*c+a*d)^(1/2)*e^(3/2)*x*(c^2*(b*x^2+a)/(a*d ^2+b*c^2)/x^2)^(1/2)*(1+a*(d*x+c)/((-a)^(1/2)*b^(1/2)*c-a*d)/x)^(1/2)*(1-a *(d*x+c)/((-a)^(1/2)*b^(1/2)*c+a*d)/x)^(1/2)*EllipticE(a^(1/2)*e^(1/2)*(d* x+c)^(1/2)/((-a)^(1/2)*b^(1/2)*c+a*d)^(1/2)/(e*x)^(1/2),(-((-a)^(1/2)*b^(1 /2)*c+a*d)/((-a)^(1/2)*b^(1/2)*c-a*d))^(1/2))/a^(1/2)/b/c/d/(b*x^2+a)^(1/2 )/(1-2*a*d*(d*x+c)/(a*d^2+b*c^2)/x+a*(d*x+c)^2/(a*d^2+b*c^2)/x^2)^(1/2)-a^ (1/2)*(b^(1/2)*c+(-a)^(1/2)*d)*((-a)^(1/2)*b^(1/2)*c+a*d)^(1/2)*e^(3/2)*x* (c^2*(b*x^2+a)/(a*d^2+b*c^2)/x^2)^(1/2)*(1+a*(d*x+c)/((-a)^(1/2)*b^(1/2)*c -a*d)/x)^(1/2)*(1-a*(d*x+c)/((-a)^(1/2)*b^(1/2)*c+a*d)/x)^(1/2)*EllipticF( a^(1/2)*e^(1/2)*(d*x+c)^(1/2)/((-a)^(1/2)*b^(1/2)*c+a*d)^(1/2)/(e*x)^(1/2) ,(-((-a)^(1/2)*b^(1/2)*c+a*d)/((-a)^(1/2)*b^(1/2)*c-a*d))^(1/2))/b^(1/2)/d /((-a)^(1/2)*b^(1/2)*c-a*d)/(b*x^2+a)^(1/2)/(1-2*a*d*(d*x+c)/(a*d^2+b*c^2) /x+a*(d*x+c)^2/(a*d^2+b*c^2)/x^2)^(1/2)-c*((-a)^(1/2)*b^(1/2)*c+a*d)^(1/2) *e^(3/2)*x*(c^2*(b*x^2+a)/(a*d^2+b*c^2)/x^2)^(1/2)*(1+a*(d*x+c)/((-a)^(1/2 )*b^(1/2)*c-a*d)/x)^(1/2)*(1-a*(d*x+c)/((-a)^(1/2)*b^(1/2)*c+a*d)/x)^(1/2) *EllipticPi(a^(1/2)*e^(1/2)*(d*x+c)^(1/2)/((-a)^(1/2)*b^(1/2)*c+a*d)^(1/2) /(e*x)^(1/2),1+a*b^(1/2)*c/(-a)^(3/2)/d,(-((-a)^(1/2)*b^(1/2)*c+a*d)/((-a) ^(1/2)*b^(1/2)*c-a*d))^(1/2))/a^(1/2)/d^2/(b*x^2+a)^(1/2)/(1-2*a*d*(d*x+c) /(a*d^2+b*c^2)/x+a*(d*x+c)^2/(a*d^2+b*c^2)/x^2)^(1/2)
Result contains complex when optimal does not.
Time = 13.54 (sec) , antiderivative size = 1263, normalized size of antiderivative = 1.22 \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:
Integrate[(e*x)^(3/2)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]),x]
Output:
(e*Sqrt[e*x]*(c + d*x)^(3/2)*(b + (b*c^2)/(c + d*x)^2 + (a*d^2)/(c + d*x)^ 2 - (2*b*c)/(c + d*x) + ((2*I)*b*c^2*(I*Sqrt[a] + Sqrt[b]*x)*Sqrt[(((I*Sqr t[b]*c)/Sqrt[a] + d)*x)/(c + d*x)]*Sqrt[(c - (I*Sqrt[a]*d)/Sqrt[b] + (I*Sq rt[b]*c*x)/Sqrt[a] + d*x)/(c + d*x)]*EllipticF[ArcSin[Sqrt[(c + (I*Sqrt[a] *d)/Sqrt[b] - (I*Sqrt[b]*c*x)/Sqrt[a] + d*x)/(2*c + 2*d*x)]], (2*Sqrt[b]*c )/(Sqrt[b]*c + I*Sqrt[a]*d)])/((I*Sqrt[b]*c + Sqrt[a]*d)*x*(c + d*x)*Sqrt[ (c + (I*Sqrt[a]*d)/Sqrt[b] - (I*Sqrt[b]*c*x)/Sqrt[a] + d*x)/(c + d*x)]) + (a*d^2*(Sqrt[a] + I*Sqrt[b]*x)*Sqrt[(((I*Sqrt[b]*c)/Sqrt[a] + d)*x)/(c + d *x)]*Sqrt[(c + (I*Sqrt[a]*d)/Sqrt[b] - (I*Sqrt[b]*c*x)/Sqrt[a] + d*x)/(c + d*x)]*(Sqrt[a]*d*EllipticE[ArcSin[Sqrt[(c + (I*Sqrt[a]*d)/Sqrt[b] - (I*Sq rt[b]*c*x)/Sqrt[a] + d*x)/(2*c + 2*d*x)]], (2*Sqrt[b]*c)/(Sqrt[b]*c + I*Sq rt[a]*d)] + ((-I)*Sqrt[b]*c - Sqrt[a]*d)*EllipticF[ArcSin[Sqrt[(c + (I*Sqr t[a]*d)/Sqrt[b] - (I*Sqrt[b]*c*x)/Sqrt[a] + d*x)/(2*c + 2*d*x)]], (2*Sqrt[ b]*c)/(Sqrt[b]*c + I*Sqrt[a]*d)]))/((b*c^2 + a*d^2)*x*(c + d*x)*Sqrt[(c - (I*Sqrt[a]*d)/Sqrt[b] + (I*Sqrt[b]*c*x)/Sqrt[a] + d*x)/(c + d*x)]) + (b*c^ 2*((-I)*Sqrt[a] + Sqrt[b]*x)*Sqrt[(((I*Sqrt[b]*c)/Sqrt[a] + d)*x)/(c + d*x )]*Sqrt[(c + (I*Sqrt[a]*d)/Sqrt[b] - (I*Sqrt[b]*c*x)/Sqrt[a] + d*x)/(c + d *x)]*(I*Sqrt[a]*d*EllipticE[ArcSin[Sqrt[(c + (I*Sqrt[a]*d)/Sqrt[b] - (I*Sq rt[b]*c*x)/Sqrt[a] + d*x)/(2*c + 2*d*x)]], (2*Sqrt[b]*c)/(Sqrt[b]*c + I*Sq rt[a]*d)] + (Sqrt[b]*c - I*Sqrt[a]*d)*EllipticF[ArcSin[Sqrt[(c + (I*Sqr...
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.
| \(\displaystyle \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 638 |
| \(\displaystyle \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x}}dx\) |
Input:
Int[(e*x)^(3/2)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]),x]
Output:
$Aborted
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Unintegrable[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x] /; FreeQ [{a, b, c, d, e, m, n, p}, x]
Time = 6.64 (sec) , antiderivative size = 595, normalized size of antiderivative = 0.57
| method | result | size |
| elliptic | \(\frac {e \sqrt {e x}\, \sqrt {e x \left (d x +c \right ) \left (b \,x^{2}+a \right )}\, \left (x \left (x -\frac {\sqrt {-a b}}{b}\right ) \left (x +\frac {\sqrt {-a b}}{b}\right )+\frac {\sqrt {-a b}\, \sqrt {-\frac {\left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) x b}{\sqrt {-a b}\, \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {-\frac {c \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{d \sqrt {-a b}\, \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {c \left (x +\frac {\sqrt {-a b}}{b}\right ) b}{d \sqrt {-a b}\, \left (x +\frac {c}{d}\right )}}\, \left (-\frac {\left (\frac {\sqrt {-a b}\, c}{b d}+\frac {c^{2}}{d^{2}}\right ) d \operatorname {EllipticF}\left (\sqrt {-\frac {\left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) x b}{\sqrt {-a b}\, \left (x +\frac {c}{d}\right )}}, \sqrt {-\frac {-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}}\right )}{\left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) c}+\frac {\sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) x b}{\sqrt {-a b}\, \left (x +\frac {c}{d}\right )}}, \sqrt {-\frac {-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}}\right ) d}{b c}+\frac {c \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) x b}{\sqrt {-a b}\, \left (x +\frac {c}{d}\right )}}, \frac {\sqrt {-a b}}{b \left (-\frac {c}{d}+\frac {\sqrt {-a b}}{b}\right )}, \sqrt {-\frac {-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}}\right )}{d \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}\right )}{b}\right )}{x \sqrt {d x +c}\, \sqrt {b \,x^{2}+a}\, \sqrt {b d e x \left (x +\frac {c}{d}\right ) \left (x -\frac {\sqrt {-a b}}{b}\right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}}\) | \(595\) |
| default | \(\text {Expression too large to display}\) | \(2601\) |
Input:
int((e*x)^(3/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
e/x*(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2)*(e*x*(d*x+c)*(b*x^2+a))^(1/2 )*(x*(x-(-a*b)^(1/2)/b)*(x+(-a*b)^(1/2)/b)+(-a*b)^(1/2)/b*(-(-(-a*b)^(1/2) /b+c/d)*x/(-a*b)^(1/2)*b/(x+c/d))^(1/2)*(x+c/d)^2*(-c/d*(x-(-a*b)^(1/2)/b) /(-a*b)^(1/2)*b/(x+c/d))^(1/2)*(c/d*(x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b/(x+c /d))^(1/2)*(-((-a*b)^(1/2)/b*c/d+c^2/d^2)/(-(-a*b)^(1/2)/b+c/d)/c*d*Ellipt icF((-(-(-a*b)^(1/2)/b+c/d)*x/(-a*b)^(1/2)*b/(x+c/d))^(1/2),(-(-c/d-(-a*b) ^(1/2)/b)/(-c/d+(-a*b)^(1/2)/b))^(1/2))+(-a*b)^(1/2)/b*EllipticE((-(-(-a*b )^(1/2)/b+c/d)*x/(-a*b)^(1/2)*b/(x+c/d))^(1/2),(-(-c/d-(-a*b)^(1/2)/b)/(-c /d+(-a*b)^(1/2)/b))^(1/2))/c*d+c/d/(-(-a*b)^(1/2)/b+c/d)*EllipticPi((-(-(- a*b)^(1/2)/b+c/d)*x/(-a*b)^(1/2)*b/(x+c/d))^(1/2),(-a*b)^(1/2)/b/(-c/d+(-a *b)^(1/2)/b),(-(-c/d-(-a*b)^(1/2)/b)/(-c/d+(-a*b)^(1/2)/b))^(1/2))))/(b*d* e*x*(x+c/d)*(x-(-a*b)^(1/2)/b)*(x+(-a*b)^(1/2)/b))^(1/2)
Timed out. \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(3/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}}}{\sqrt {a + b x^{2}} \sqrt {c + d x}}\, dx \] Input:
integrate((e*x)**(3/2)/(d*x+c)**(1/2)/(b*x**2+a)**(1/2),x)
Output:
Integral((e*x)**(3/2)/(sqrt(a + b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{\sqrt {b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate((e*x)^(3/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x)^(3/2)/(sqrt(b*x^2 + a)*sqrt(d*x + c)), x)
\[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{\sqrt {b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate((e*x)^(3/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x)^(3/2)/(sqrt(b*x^2 + a)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{\sqrt {b\,x^2+a}\,\sqrt {c+d\,x}} \,d x \] Input:
int((e*x)^(3/2)/((a + b*x^2)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int((e*x)^(3/2)/((a + b*x^2)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {(e x)^{3/2}}{\sqrt {c+d x} \sqrt {a+b x^2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b \,x^{2}+a}\, x}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) e \] Input:
int((e*x)^(3/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2),x)
Output:
sqrt(e)*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x**2)*x)/(a*c + a*d*x + b*c* x**2 + b*d*x**3),x)*e