Integrand size = 32, antiderivative size = 280 \[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=-\frac {2 A \sqrt {a-b x^2}}{a c e \sqrt {e x}}-\frac {2 A \sqrt [4]{b} \sqrt {\frac {a-b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt [4]{a} c e^{3/2} \sqrt {a-b x^2}}+\frac {2 A \sqrt [4]{b} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{a} c e^{3/2} \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} (B c-A d) \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2}} \] Output:
-2*A*(-b*x^2+a)^(1/2)/a/c/e/(e*x)^(1/2)-2*A*b^(1/4)*((-b*x^2+a)/a)^(1/2)*E llipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/a^(1/4)/c/e^(3/2)/(-b*x^2+ a)^(1/2)+2*A*b^(1/4)*((-b*x^2+a)/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^ (1/4)/e^(1/2),I)/a^(1/4)/c/e^(3/2)/(-b*x^2+a)^(1/2)+2*a^(1/4)*(-A*d+B*c)*( (-b*x^2+a)/a)^(1/2)*EllipticPi(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),-a^(1/2 )*d/b^(1/2)/c,I)/b^(1/4)/c^2/e^(3/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.76 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=\frac {2 i \sqrt {1-\frac {a}{b x^2}} x^{5/2} \left (A \sqrt {b} c E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )-\left (A \sqrt {b} c-\sqrt {a} B c+\sqrt {a} A d\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+\sqrt {a} (-B c+A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )}{\sqrt {a} \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c^2 (e x)^{3/2} \sqrt {a-b x^2}} \] Input:
Integrate[(A + B*x)/((e*x)^(3/2)*(c + d*x)*Sqrt[a - b*x^2]),x]
Output:
((2*I)*Sqrt[1 - a/(b*x^2)]*x^(5/2)*(A*Sqrt[b]*c*EllipticE[I*ArcSinh[Sqrt[- (Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] - (A*Sqrt[b]*c - Sqrt[a]*B*c + Sqrt[a]*A* d)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] + Sqrt[a]*(- (B*c) + A*d)*EllipticPi[-((Sqrt[b]*c)/(Sqrt[a]*d)), I*ArcSinh[Sqrt[-(Sqrt[ a]/Sqrt[b])]/Sqrt[x]], -1]))/(Sqrt[a]*Sqrt[-(Sqrt[a]/Sqrt[b])]*c^2*(e*x)^( 3/2)*Sqrt[a - b*x^2])
Time = 1.42 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2354, 2249, 2009}
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.
| \(\displaystyle \int \frac {A+B x}{(e x)^{3/2} \sqrt {a-b x^2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2354 |
| \(\displaystyle \frac {2 \int \frac {A e+B x e}{e x (c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \frac {2 \int \left (\frac {A}{c e x \sqrt {a-b x^2}}+\frac {B c-A d}{c (c e+d x e) \sqrt {a-b x^2}}\right )d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 \sqrt {e} \sqrt {a-b x^2}}+\frac {A \sqrt [4]{b} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{a} c \sqrt {e} \sqrt {a-b x^2}}-\frac {A \sqrt [4]{b} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt [4]{a} c \sqrt {e} \sqrt {a-b x^2}}-\frac {A \sqrt {a-b x^2}}{a c \sqrt {e x}}\right )}{e}\) |
Input:
Int[(A + B*x)/((e*x)^(3/2)*(c + d*x)*Sqrt[a - b*x^2]),x]
Output:
(2*(-((A*Sqrt[a - b*x^2])/(a*c*Sqrt[e*x])) - (A*b^(1/4)*Sqrt[1 - (b*x^2)/a ]*EllipticE[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(a^(1/4)*c *Sqrt[e]*Sqrt[a - b*x^2]) + (A*b^(1/4)*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSi n[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(a^(1/4)*c*Sqrt[e]*Sqrt[a - b*x^2]) + (a^(1/4)*(B*c - A*d)*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a]* d)/(Sqrt[b]*c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1 /4)*c^2*Sqrt[e]*Sqrt[a - b*x^2])))/e
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & & PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Int[(Px_)*((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2) ^(p_.), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[(Px /. x -> x^k/e)*x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x ], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolyQ[Px, x] && FractionQ[m]
Time = 2.23 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(-\frac {2 A \sqrt {-b \,x^{2}+a}}{a c e \sqrt {e x}}-\frac {\left (\frac {A \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {-b e \,x^{3}+a e x}}+\frac {a \left (A d -B c \right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right ) \sqrt {x e \left (-b \,x^{2}+a \right )}}{c a e \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(375\) |
| elliptic | \(\frac {\sqrt {x e \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b e \,x^{2}+a e \right ) A}{e^{2} a c \sqrt {x \left (-b e \,x^{2}+a e \right )}}-\frac {A \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{a e c \sqrt {-b e \,x^{3}+a e x}}-\frac {\left (A d -B c \right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{e c d b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(391\) |
| default | \(\frac {2 A \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b c \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-2 A \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a d \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {a b}-A \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b c \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+A \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a d \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {a b}-A \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{\sqrt {a b}\, d -b c}, \frac {\sqrt {2}}{2}\right ) a d \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {a b}+B \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{\sqrt {a b}\, d -b c}, \frac {\sqrt {2}}{2}\right ) a c \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {a b}+2 A \,b^{2} c \,x^{2}-2 A b d \,x^{2} \sqrt {a b}-2 A a b c +2 A a d \sqrt {a b}}{\sqrt {-b \,x^{2}+a}\, c \left (-\sqrt {a b}\, d +b c \right ) a e \sqrt {e x}}\) | \(604\) |
Input:
int((B*x+A)/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*A*(-b*x^2+a)^(1/2)/a/c/e/(e*x)^(1/2)-1/c/a*(A*(a*b)^(1/2)*((x+1/b*(a*b) ^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)* (-b/(a*b)^(1/2)*x)^(1/2)/(-b*e*x^3+a*e*x)^(1/2)*(-2/b*(a*b)^(1/2)*Elliptic E(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/b*(a*b)^(1/2)*E llipticF(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2)))+a*(A*d-B* c)/d/b*(a*b)^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a *b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-b/(a*b)^(1/2)*x)^(1/2)/(-b*e*x^3+a*e*x)^ (1/2)/(c/d-1/b*(a*b)^(1/2))*EllipticPi(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2)) ^(1/2),-1/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)),1/2*2^(1/2)))/e*(x*e*(-b*x^2 +a))^(1/2)/(e*x)^(1/2)/(-b*x^2+a)^(1/2)
Timed out. \[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="frica s")
Output:
Timed out
\[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=\int \frac {A + B x}{\left (e x\right )^{\frac {3}{2}} \sqrt {a - b x^{2}} \left (c + d x\right )}\, dx \] Input:
integrate((B*x+A)/(e*x)**(3/2)/(d*x+c)/(-b*x**2+a)**(1/2),x)
Output:
Integral((A + B*x)/((e*x)**(3/2)*sqrt(a - b*x**2)*(c + d*x)), x)
\[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=\int { \frac {B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="maxim a")
Output:
integrate((B*x + A)/(sqrt(-b*x^2 + a)*(d*x + c)*(e*x)^(3/2)), x)
\[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=\int { \frac {B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="giac" )
Output:
integrate((B*x + A)/(sqrt(-b*x^2 + a)*(d*x + c)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{3/2}\,\sqrt {a-b\,x^2}\,\left (c+d\,x\right )} \,d x \] Input:
int((A + B*x)/((e*x)^(3/2)*(a - b*x^2)^(1/2)*(c + d*x)),x)
Output:
int((A + B*x)/((e*x)^(3/2)*(a - b*x^2)^(1/2)*(c + d*x)), x)
\[ \int \frac {A+B x}{(e x)^{3/2} (c+d x) \sqrt {a-b x^2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {x}\, \sqrt {-b \,x^{2}+a}-\left (\int \frac {\sqrt {-b \,x^{2}+a}}{\sqrt {x}\, a c +\sqrt {x}\, a d x -\sqrt {x}\, b c \,x^{2}-\sqrt {x}\, b d \,x^{3}}d x \right ) a d x -\left (\int \frac {\sqrt {-b \,x^{2}+a}\, x^{2}}{\sqrt {x}\, a c +\sqrt {x}\, a d x -\sqrt {x}\, b c \,x^{2}-\sqrt {x}\, b d \,x^{3}}d x \right ) b d x -\left (\int \frac {\sqrt {-b \,x^{2}+a}\, x}{\sqrt {x}\, a c +\sqrt {x}\, a d x -\sqrt {x}\, b c \,x^{2}-\sqrt {x}\, b d \,x^{3}}d x \right ) b c x +\left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) b c x \right )}{c \,e^{2} x} \] Input:
int((B*x+A)/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x)
Output:
(sqrt(e)*( - 2*sqrt(x)*sqrt(a - b*x**2) - int(sqrt(a - b*x**2)/(sqrt(x)*a* c + sqrt(x)*a*d*x - sqrt(x)*b*c*x**2 - sqrt(x)*b*d*x**3),x)*a*d*x - int((s qrt(a - b*x**2)*x**2)/(sqrt(x)*a*c + sqrt(x)*a*d*x - sqrt(x)*b*c*x**2 - sq rt(x)*b*d*x**3),x)*b*d*x - int((sqrt(a - b*x**2)*x)/(sqrt(x)*a*c + sqrt(x) *a*d*x - sqrt(x)*b*c*x**2 - sqrt(x)*b*d*x**3),x)*b*c*x + int((sqrt(x)*sqrt (a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x**3 - b*d*x**4),x)*b*c*x))/(c*e**2* x)