Integrand size = 26, antiderivative size = 322 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=\frac {2 \sqrt {c+d x}}{a e (e x)^{3/2} \sqrt {a+b x}}+\frac {2 (8 b c-a d) \sqrt {c+d x}}{3 a^2 c e^2 \sqrt {e x} \sqrt {a+b x}}-\frac {8 \sqrt {a+b x} \sqrt {c+d x}}{3 a^2 e (e x)^{3/2}}+\frac {2 \sqrt {b} (8 b c-a d) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{3 a^{5/2} c e^{5/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {8 \sqrt {b} d \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{3 a^{3/2} c e^{5/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:
2*(d*x+c)^(1/2)/a/e/(e*x)^(3/2)/(b*x+a)^(1/2)+2/3*(-a*d+8*b*c)*(d*x+c)^(1/ 2)/a^2/c/e^2/(e*x)^(1/2)/(b*x+a)^(1/2)-8/3*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2 /e/(e*x)^(3/2)+2/3*b^(1/2)*(-a*d+8*b*c)*(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))/a^(5/2)/c/e^( 5/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)-8/3*b^(1/2)*d*(d*x+c)^(1/2) *InverseJacobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)),(1-a*d/b/c)^( 1/2))/a^(3/2)/c/e^(5/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)
Result contains complex when optimal does not.
Time = 9.74 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=\frac {x \left (-2 a c (a+4 b x) (c+d x)+2 i \sqrt {\frac {a}{b}} b d (-8 b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} b d (-4 b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{3 a^3 c (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:
Integrate[Sqrt[c + d*x]/((e*x)^(5/2)*(a + b*x)^(3/2)),x]
Output:
(x*(-2*a*c*(a + 4*b*x)*(c + d*x) + (2*I)*Sqrt[a/b]*b*d*(-8*b*c + a*d)*Sqrt [1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt [x]], (b*c)/(a*d)] - (2*I)*Sqrt[a/b]*b*d*(-4*b*c + a*d)*Sqrt[1 + a/(b*x)]* Sqrt[1 + c/(d*x)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a *d)]))/(3*a^3*c*(e*x)^(5/2)*Sqrt[a + b*x]*Sqrt[c + d*x])
Time = 0.44 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {110, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \int -\frac {4 b c-a d+3 b d x}{2 (e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 b c-a d+3 b d x}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {b e (c (8 b c-5 a d)+d (4 b c-a d) x)}{2 \sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e^2}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {b \int \frac {c (8 b c-5 a d)+d (4 b c-a d) x}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \int -\frac {d (b c-a d) e (4 a c+(8 b c-a d) x)}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a e (b c-a d)}+\frac {2 \sqrt {e x} \sqrt {c+d x} (8 b c-a d)}{a e \sqrt {a+b x}}\right )}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} (8 b c-a d)}{a e \sqrt {a+b x}}-\frac {d \int \frac {4 a c+(8 b c-a d) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} (8 b c-a d)}{a e \sqrt {a+b x}}-\frac {d \left (\frac {(8 b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {c (8 b c-5 a d) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} (8 b c-a d)}{a e \sqrt {a+b x}}-\frac {d \left (\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (8 b c-a d) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (8 b c-5 a d) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} (8 b c-a d)}{a e \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (8 b c-a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (8 b c-5 a d) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} (8 b c-a d)}{a e \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (8 b c-a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (8 b c-5 a d) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} (8 b c-a d)}{a e \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (8 b c-a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (8 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} (4 b c-a d)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a e}-\frac {2 \sqrt {c+d x}}{3 a e (e x)^{3/2} \sqrt {a+b x}}\) |
Input:
Int[Sqrt[c + d*x]/((e*x)^(5/2)*(a + b*x)^(3/2)),x]
Output:
(-2*Sqrt[c + d*x])/(3*a*e*(e*x)^(3/2)*Sqrt[a + b*x]) - ((-2*(4*b*c - a*d)* Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]*Sqrt[a + b*x]) - (b*((2*(8*b*c - a*d)*Sqrt [e*x]*Sqrt[c + d*x])/(a*e*Sqrt[a + b*x]) - (d*((2*Sqrt[-a]*(8*b*c - a*d)*S qrt[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*c - 5*a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c] *EllipticF[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/( Sqrt[b]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[c + d*x])))/a))/(a*c*e))/(3*a*e)
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 = 3.31 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.55
| method | result | size |
| default | \(-\frac {2 \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2} x -4 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d x -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2} x +9 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d x -8 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3} x +a b \,d^{3} x^{3}-8 b^{2} c \,d^{2} x^{3}+a^{2} d^{3} x^{2}-3 a b c \,d^{2} x^{2}-8 b^{2} c^{2} d \,x^{2}+2 a^{2} c \,d^{2} x -4 a b \,c^{2} d x +a^{2} c^{2} d \right )}{3 x c \,e^{2} \sqrt {e x}\, d \,a^{3} \sqrt {b x +a}\, \sqrt {x d +c}}\) | \(500\) |
| elliptic | \(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{3 e^{3} a^{2} x^{2}}-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right ) \left (a d -5 b c \right )}{3 e^{3} a^{3} c \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (b d e \,x^{2}+b c e x \right ) b}{a^{3} e^{3} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (-\frac {d b}{3 a^{2} e^{2}}-\frac {\left (a d +b c \right ) \left (a d -5 b c \right )}{3 a^{3} c \,e^{2}}+\frac {\left (a d e +b c e \right ) \left (a d -5 b c \right )}{3 e^{3} a^{3} c}-\frac {\left (a d -b c \right ) b}{a^{3} e^{2}}-\frac {b^{2} c}{e^{2} a^{3}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (\frac {d b \left (a d -5 b c \right )}{3 a^{3} c \,e^{2}}-\frac {d \,b^{2}}{a^{3} e^{2}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\) | \(612\) |
Input:
int((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-1/c*x*d)^(1/2)*Ellip ticF(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*a^2*c*d^2*x-4*((d*x+c)/c)^( 1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-1/c*x*d)^(1/2)*EllipticF(((d*x+c)/c)^(1 /2),(-b*c/(a*d-b*c))^(1/2))*a*b*c^2*d*x-((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d- b*c))^(1/2)*(-1/c*x*d)^(1/2)*EllipticE(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^ (1/2))*a^2*c*d^2*x+9*((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-1/c*x *d)^(1/2)*EllipticE(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*a*b*c^2*d*x- 8*((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-1/c*x*d)^(1/2)*EllipticE (((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*b^2*c^3*x+a*b*d^3*x^3-8*b^2*c*d ^2*x^3+a^2*d^3*x^2-3*a*b*c*d^2*x^2-8*b^2*c^2*d*x^2+2*a^2*c*d^2*x-4*a*b*c^2 *d*x+a^2*c^2*d)/x/c/e^2/(e*x)^(1/2)/d/a^3/(b*x+a)^(1/2)/(d*x+c)^(1/2)
Time = 0.09 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (a^{2} b c d - {\left (8 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} - {\left (4 \, a b^{2} c d - a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} - {\left ({\left (8 \, b^{3} c^{2} - 5 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{3} + {\left (8 \, a b^{2} c^{2} - 5 \, a^{2} b c d - a^{3} d^{2}\right )} x^{2}\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) - 3 \, {\left ({\left (8 \, b^{3} c d - a b^{2} d^{2}\right )} x^{3} + {\left (8 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{9 \, {\left (a^{3} b^{2} c d e^{3} x^{3} + a^{4} b c d e^{3} x^{2}\right )}} \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x+a)^(3/2),x, algorithm="fricas")
Output:
-2/9*(3*(a^2*b*c*d - (8*b^3*c*d - a*b^2*d^2)*x^2 - (4*a*b^2*c*d - a^2*b*d^ 2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x) - ((8*b^3*c^2 - 5*a*b^2*c*d - a^2*b*d^2)*x^3 + (8*a*b^2*c^2 - 5*a^2*b*c*d - a^3*d^2)*x^2)*sqrt(b*d*e)*we ierstrassPInverse(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*((8*b^3*c*d - a*b^2*d^2)*x^3 + (8*a*b^2*c*d - a^2 *b*d^2)*x^2)*sqrt(b*d*e)*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))))/(a^3*b^2*c*d*e^3*x^3 + a^4*b*c*d*e^3*x ^2)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=\int \frac {\sqrt {c + d x}}{\left (e x\right )^{\frac {5}{2}} \left (a + b x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(1/2)/(e*x)**(5/2)/(b*x+a)**(3/2),x)
Output:
Integral(sqrt(c + d*x)/((e*x)**(5/2)*(a + b*x)**(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)/((b*x + a)^(3/2)*(e*x)^(5/2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)/((b*x + a)^(3/2)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=\int \frac {\sqrt {c+d\,x}}{{\left (e\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(1/2)/((e*x)^(5/2)*(a + b*x)^(3/2)),x)
Output:
int((c + d*x)^(1/2)/((e*x)^(5/2)*(a + b*x)^(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x)^{3/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a^{2} x^{2}+2 \sqrt {x}\, a b \,x^{3}+\sqrt {x}\, b^{2} x^{4}}d x \right )}{e^{3}} \] Input:
int((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x+a)^(3/2),x)
Output:
(sqrt(e)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*x**2 + 2*sqrt(x)* a*b*x**3 + sqrt(x)*b**2*x**4),x))/e**3