Integrand size = 26, antiderivative size = 359 \[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \left (9 a b c d-8 (b c+a d)^2\right ) e^2 \sqrt {e x} \sqrt {c+d x}}{15 b^2 d^3 \sqrt {a+b x}}-\frac {8 (b c+a d) e^2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{15 b^2 d^2}+\frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}+\frac {2 \sqrt {a} \left (9 a b c d-8 (b c+a d)^2\right ) e^{5/2} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {8 a^{3/2} (b c+a d) e^{5/2} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} d^2 \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:
-2/15*(9*a*b*c*d-8*(a*d+b*c)^2)*e^2*(e*x)^(1/2)*(d*x+c)^(1/2)/b^2/d^3/(b*x +a)^(1/2)-8/15*(a*d+b*c)*e^2*(e*x)^(1/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d ^2+2/5*e*(e*x)^(3/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b/d+2/15*a^(1/2)*(9*a*b*c *d-8*(a*d+b*c)^2)*e^(5/2)*(d*x+c)^(1/2)*EllipticE(b^(1/2)*(e*x)^(1/2)/a^(1 /2)/e^(1/2)/(1+b*x/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(5/2)/d^3/(b*x+a)^(1/2)/( a*(d*x+c)/c/(b*x+a))^(1/2)+8/15*a^(3/2)*(a*d+b*c)*e^(5/2)*(d*x+c)^(1/2)*In verseJacobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)),(1-a*d/b/c)^(1/2 ))/b^(5/2)/d^2/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)
Result contains complex when optimal does not.
Time = 12.76 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.82 \[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 e^3 \left ((a+b x) (c+d x) \left (8 a^2 d^2+a b d (7 c-4 d x)+b^2 \left (8 c^2-4 c d x+3 d^2 x^2\right )\right )+i \sqrt {\frac {a}{b}} b d \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-i \sqrt {\frac {a}{b}} b d \left (4 b^2 c^2+3 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{15 b^3 d^3 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:
Integrate[(e*x)^(5/2)/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]
Output:
(2*e^3*((a + b*x)*(c + d*x)*(8*a^2*d^2 + a*b*d*(7*c - 4*d*x) + b^2*(8*c^2 - 4*c*d*x + 3*d^2*x^2)) + I*Sqrt[a/b]*b*d*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d ^2)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[a /b]/Sqrt[x]], (b*c)/(a*d)] - I*Sqrt[a/b]*b*d*(4*b^2*c^2 + 3*a*b*c*d + 8*a^ 2*d^2)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqr t[a/b]/Sqrt[x]], (b*c)/(a*d)]))/(15*b^3*d^3*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x])
Time = 0.41 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {113, 27, 171, 27, 176, 122, 120, 127, 126}
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.
| \(\displaystyle \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 113 |
| \(\displaystyle \frac {2 \int -\frac {e^2 \sqrt {e x} (3 a c+4 (b c+a d) x)}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{5 b d}+\frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \int \frac {\sqrt {e x} (3 a c+4 (b c+a d) x)}{\sqrt {a+b x} \sqrt {c+d x}}dx}{5 b d}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \left (\frac {2 \int -\frac {e \left (4 a c (b c+a d)-\left (9 a b c d-8 (b c+a d)^2\right ) x\right )}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}+\frac {8 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 b d}\right )}{5 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \left (\frac {8 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 b d}-\frac {e \int \frac {4 a c (b c+a d)-\left (9 a b c d-8 (b c+a d)^2\right ) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}\right )}{5 b d}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \left (\frac {8 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 b d}-\frac {e \left (-\frac {c \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}\right )}{3 b d}\right )}{5 b d}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \left (\frac {8 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 b d}-\frac {e \left (-\frac {c \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b d}\right )}{5 b d}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \left (\frac {8 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 b d}-\frac {e \left (-\frac {c \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (9 a b c d-8 (a d+b c)^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b d}\right )}{5 b d}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \left (\frac {8 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 b d}-\frac {e \left (-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (9 a b c d-8 (a d+b c)^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b d}\right )}{5 b d}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {2 e (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}{5 b d}-\frac {e^2 \left (\frac {8 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 b d}-\frac {e \left (-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (9 a b c d-8 (a d+b c)^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b d}\right )}{5 b d}\) |
Input:
Int[(e*x)^(5/2)/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]
Output:
(2*e*(e*x)^(3/2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(5*b*d) - (e^2*((8*(b*c + a* d)*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x])/(3*b*d) - (e*((-2*Sqrt[-a]*(9*a* b*c*d - 8*(b*c + a*d)^2)*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[ (Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[e]* Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (2*Sqrt[-a]*c*(8*b^2*c^2 + 3*a*b*c*d + 4*a^2*d^2)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*S qrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[e]*Sqrt[a + b *x]*Sqrt[c + d*x])))/(3*b*d)))/(5*b*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 1.26 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.37
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 e^{2} x \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{5 b d}-\frac {4 e \left (2 a d e +2 b c e \right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{15 b^{2} d^{2}}+\frac {4 e^{2} \left (2 a d e +2 b c e \right ) a \,c^{2} \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{15 b^{2} d^{3} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (-\frac {3 e^{3} a c}{5 b d}+\frac {4 e \left (2 a d e +2 b c e \right ) \left (a d e +b c e \right )}{15 b^{2} d^{2}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{e x \sqrt {x d +c}\, \sqrt {b x +a}}\) | \(491\) |
| default | \(-\frac {2 \left (8 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}+3 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}+4 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d -8 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}+\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}-\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d +8 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{3} c^{4}-3 b^{3} d^{4} x^{4}+a \,b^{2} d^{4} x^{3}+b^{3} c \,d^{3} x^{3}+4 a^{2} b \,d^{4} x^{2}+5 a \,b^{2} c \,d^{3} x^{2}+4 b^{3} c^{2} d^{2} x^{2}+4 a^{2} b c \,d^{3} x +4 a \,b^{2} c^{2} d^{2} x \right ) \sqrt {x d +c}\, \sqrt {b x +a}\, e^{2} \sqrt {e x}}{15 x \,d^{4} b^{3} \left (b d \,x^{2}+a d x +b c x +a c \right )}\) | \(689\) |
Input:
int((e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/e/x*(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*(e*x*(b*x+a)*(d*x+c))^(1/2)* (2/5*e^2/b/d*x*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)-4/15*e/b^2/d^ 2*(2*a*d*e+2*b*c*e)*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)+4/15*e^2 /b^2/d^3*(2*a*d*e+2*b*c*e)*a*c^2*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^ (1/2)*(-1/c*x*d)^(1/2)/(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)*Ellip ticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2*(-3/5*e^3/b/d*a*c+4/15 *e/b^2/d^2*(2*a*d*e+2*b*c*e)*(a*d*e+b*c*e))*c/d*((x+c/d)/c*d)^(1/2)*((x+a/ b)/(-c/d+a/b))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e *x)^(1/2)*((-c/d+a/b)*EllipticE(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2 ))-a/b*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))))
Time = 0.11 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.16 \[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left ({\left (8 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3}\right )} \sqrt {b d e} e^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (8 \, b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} \sqrt {b d e} e^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) - 3 \, {\left (3 \, b^{3} d^{3} e^{2} x - 4 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}\right )}}{45 \, b^{4} d^{4}} \] Input:
integrate((e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
-2/45*((8*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 8*a^3*d^3)*sqrt(b*d*e) *e^2*weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/2 7*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*( 3*b*d*x + b*c + a*d)/(b*d)) + 3*(8*b^3*c^2*d + 7*a*b^2*c*d^2 + 8*a^2*b*d^3 )*sqrt(b*d*e)*e^2*weierstrassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d ^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^ 3), weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27 *(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3 *b*d*x + b*c + a*d)/(b*d))) - 3*(3*b^3*d^3*e^2*x - 4*(b^3*c*d^2 + a*b^2*d^ 3)*e^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x))/(b^4*d^4)
\[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {\left (e x\right )^{\frac {5}{2}}}{\sqrt {a + b x} \sqrt {c + d x}}\, dx \] Input:
integrate((e*x)**(5/2)/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
Output:
Integral((e*x)**(5/2)/(sqrt(a + b*x)*sqrt(c + d*x)), x)
\[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{2}}}{\sqrt {b x + a} \sqrt {d x + c}} \,d x } \] Input:
integrate((e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x)^(5/2)/(sqrt(b*x + a)*sqrt(d*x + c)), x)
\[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{2}}}{\sqrt {b x + a} \sqrt {d x + c}} \,d x } \] Input:
integrate((e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate((e*x)^(5/2)/(sqrt(b*x + a)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}}{\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \] Input:
int((e*x)^(5/2)/((a + b*x)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int((e*x)^(5/2)/((a + b*x)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {(e x)^{5/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {e}\, e^{2} \left (-6 \sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, a c +4 \sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, a d x +4 \sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, b c x -8 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a^{3} d^{3}-15 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a^{2} b c \,d^{2}-15 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a \,b^{2} c^{2} d -8 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) b^{3} c^{3}+3 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{a b \,d^{2} x^{3}+b^{2} c d \,x^{3}+a^{2} d^{2} x^{2}+2 a b c d \,x^{2}+b^{2} c^{2} x^{2}+a^{2} c d x +a b \,c^{2} x}d x \right ) a^{3} c^{2} d +3 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{a b \,d^{2} x^{3}+b^{2} c d \,x^{3}+a^{2} d^{2} x^{2}+2 a b c d \,x^{2}+b^{2} c^{2} x^{2}+a^{2} c d x +a b \,c^{2} x}d x \right ) a^{2} b \,c^{3}\right )}{10 b d \left (a d +b c \right )} \] Input:
int((e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x)
Output:
(sqrt(e)*e**2*( - 6*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*a*c + 4*sqrt(x)*sq rt(c + d*x)*sqrt(a + b*x)*a*d*x + 4*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*b* c*x - 8*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2* x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2), x)*a**3*d**3 - 15*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2* c*d*x**2),x)*a**2*b*c*d**2 - 15*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x )/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2* c**2*x + b**2*c*d*x**2),x)*a*b**2*c**2*d - 8*int((sqrt(x)*sqrt(c + d*x)*sq rt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2 *x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*b**3*c**3 + 3*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a**2*c*d*x + a**2*d**2*x**2 + a*b*c**2*x + 2*a*b*c* d*x**2 + a*b*d**2*x**3 + b**2*c**2*x**2 + b**2*c*d*x**3),x)*a**3*c**2*d + 3*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a**2*c*d*x + a**2*d**2*x**2 + a*b*c**2*x + 2*a*b*c*d*x**2 + a*b*d**2*x**3 + b**2*c**2*x**2 + b**2*c*d*x **3),x)*a**2*b*c**3))/(10*b*d*(a*d + b*c))