Integrand size = 26, antiderivative size = 334 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}+\frac {4 (2 b c+a d) \sqrt {c+d x}}{3 a^2 c^2 e^2 \sqrt {e x} \sqrt {a+b x}}+\frac {2 \sqrt {b} \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{3 a^{5/2} c^2 (b c-a d) e^{5/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {2 \sqrt {b} d (4 b c-a d) \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 (b c-a d) e^{5/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:
-2/3*(d*x+c)^(1/2)/a/c/e/(e*x)^(3/2)/(b*x+a)^(1/2)+4/3*(a*d+2*b*c)*(d*x+c) ^(1/2)/a^2/c^2/e^2/(e*x)^(1/2)/(b*x+a)^(1/2)+2/3*b^(1/2)*(-2*a^2*d^2-3*a*b *c*d+8*b^2*c^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))/a^(5/2)/c^2/(-a*d+b*c)/e^(5/2)/(b*x+a )^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)-2/3*b^(1/2)*d*(-a*d+4*b*c)*(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^2/(-a*d+b*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 = 7.38 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {x \left (-2 a c (c+d x) \left (a^2 d-4 b^2 c x+a b (-c+d x)\right )-2 i \sqrt {\frac {a}{b}} b d \left (-8 b^2 c^2+3 a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+4 i \sqrt {\frac {a}{b}} b d \left (-2 b^2 c^2+a b c d+a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{3 a^3 c^2 (-b c+a d) (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:
Integrate[1/((e*x)^(5/2)*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
Output:
(x*(-2*a*c*(c + d*x)*(a^2*d - 4*b^2*c*x + a*b*(-c + d*x)) - (2*I)*Sqrt[a/b ]*b*d*(-8*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*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)] + (4*I)* Sqrt[a/b]*b*d*(-2*b^2*c^2 + a*b*c*d + a^2*d^2)*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^2*(-(b*c) + a*d)*(e*x)^(5/2)*Sqrt[a + b*x]*Sqrt[c + d*x])
Time = 0.48 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.30, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {115, 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.
| \(\displaystyle \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle -\frac {2 \int \frac {e (2 (2 b c+a d)+3 b d x)}{2 (e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c e^2}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {2 (2 b c+a d)+3 b d x}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {b e (c (8 b c+a d)+2 d (2 b c+a d) x)}{2 \sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e^2}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {b \int \frac {c (8 b c+a d)+2 d (2 b c+a d) x}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \int -\frac {d e \left (a c (4 b c-a d)+\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) x\right )}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a e (b c-a d)}+\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a e \sqrt {a+b x} (b c-a d)}\right )}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \int \frac {a c (4 b c-a d)+\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}\right )}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {\left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {c (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a (b c-a d)}\right )}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {-\frac {b \left (\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a e \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) (a d+8 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{a (b c-a d)}\right )}{a c e}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2} \sqrt {a+b x}}\) |
Input:
Int[1/((e*x)^(5/2)*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
Output:
(-2*Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)*Sqrt[a + b*x]) - ((-4*(2*b*c + a*d )*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]*Sqrt[a + b*x]) - (b*((2*(8*b^2*c^2 - 3*a *b*c*d - 2*a^2*d^2)*Sqrt[e*x]*Sqrt[c + d*x])/(a*(b*c - a*d)*e*Sqrt[a + b*x ]) - (d*((2*Sqrt[-a]*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*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*(b*c - a*d)*(8*b*c + a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*Ellip ticF[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*(b*c - a*d))))/(a*c*e))/(3*a *c*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[b*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && 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[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(644\) vs. \(2(287)=574\).
Time = 4.65 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.93
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{3 e^{3} a^{2} c \,x^{2}}+\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right ) \left (2 a d +5 b c \right )}{3 e^{3} a^{3} c^{2} \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}-\frac {2 \left (b d e \,x^{2}+b c e x \right ) b^{2}}{\left (a d -b c \right ) a^{3} e^{3} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (-\frac {d b}{3 a^{2} c \,e^{2}}+\frac {\left (a d +b c \right ) \left (2 a d +5 b c \right )}{3 a^{3} c^{2} e^{2}}-\frac {\left (a d e +b c e \right ) \left (2 a d +5 b c \right )}{3 e^{3} a^{3} c^{2}}+\frac {b^{2}}{a^{3} e^{2}}+\frac {b^{3} c}{e^{2} \left (a d -b c \right ) a^{3}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (-\frac {d b \left (2 a d +5 b c \right )}{3 a^{3} c^{2} e^{2}}+\frac {b^{3} d}{\left (a d -b c \right ) a^{3} e^{2}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\) | \(645\) |
| default | \(\frac {\frac {4 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 {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}}{3}+\frac {4 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 {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}}{3}-\frac {8 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 {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d}{3}-\frac {4 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^{3} c \,d^{3}}{3}-\frac {2 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} b \,c^{2} d^{2}}{3}+\frac {22 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 \,b^{2} c^{3} d}{3}-\frac {16 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 ) b^{3} c^{4}}{3}+\frac {4 x^{3} a^{2} b \,d^{4}}{3}+2 x^{3} a \,b^{2} c \,d^{3}-\frac {16 x^{3} b^{3} c^{2} d^{2}}{3}+\frac {4 x^{2} a^{3} d^{4}}{3}+\frac {8 x^{2} a^{2} b c \,d^{3}}{3}-\frac {2 x^{2} a \,b^{2} c^{2} d^{2}}{3}-\frac {16 x^{2} b^{3} c^{3} d}{3}+\frac {2 x \,a^{3} c \,d^{3}}{3}+2 x \,a^{2} b \,c^{2} d^{2}-\frac {8 x a \,b^{2} c^{3} d}{3}-\frac {2 a^{3} c^{2} d^{2}}{3}+\frac {2 a^{2} c^{3} d b}{3}}{x \,e^{2} \sqrt {e x}\, \left (a d -b c \right ) d \,a^{3} c^{2} \sqrt {x d +c}\, \sqrt {b x +a}}\) | \(737\) |
Input:
int(1/(e*x)^(5/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
(e*x*(b*x+a)*(d*x+c))^(1/2)/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*(-2/3/ e^3/a^2/c*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^2+2/3*(b*d*e*x^2 +a*d*e*x+b*c*e*x+a*c*e)/e^3/a^3/c^2*(2*a*d+5*b*c)/(x*(b*d*e*x^2+a*d*e*x+b* c*e*x+a*c*e))^(1/2)-2*(b*d*e*x^2+b*c*e*x)/(a*d-b*c)*b^2/a^3/e^3/((x+a/b)*( b*d*e*x^2+b*c*e*x))^(1/2)+2*(-1/3*d*b/a^2/c/e^2+1/3*(a*d+b*c)*(2*a*d+5*b*c )/a^3/c^2/e^2-1/3*(a*d*e+b*c*e)/e^3/a^3/c^2*(2*a*d+5*b*c)+b^2/a^3/e^2+b^3* c/e^2/(a*d-b*c)/a^3)*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)*EllipticF(((x +c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2*(-1/3*d*b*(2*a*d+5*b*c)/a^3/c^ 2/e^2+b^3*d/(a*d-b*c)/a^3/e^2)*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*Ellipt icF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (287) = 574\).
Time = 0.10 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (3 \, {\left (a^{2} b^{2} c^{2} d - a^{3} b c d^{2} - {\left (8 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} x^{2} - 2 \, {\left (2 \, a b^{3} c^{2} d - a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} - {\left ({\left (8 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{3} + {\left (8 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d - 2 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{2}\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) - 3 \, {\left ({\left (8 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} x^{3} + {\left (8 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{9 \, {\left ({\left (a^{3} b^{3} c^{3} d - a^{4} b^{2} c^{2} d^{2}\right )} e^{3} x^{3} + {\left (a^{4} b^{2} c^{3} d - a^{5} b c^{2} d^{2}\right )} e^{3} x^{2}\right )}} \] Input:
integrate(1/(e*x)^(5/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
-2/9*(3*(a^2*b^2*c^2*d - a^3*b*c*d^2 - (8*b^4*c^2*d - 3*a*b^3*c*d^2 - 2*a^ 2*b^2*d^3)*x^2 - 2*(2*a*b^3*c^2*d - a^2*b^2*c*d^2 - a^3*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x) - ((8*b^4*c^3 - 7*a*b^3*c^2*d - 2*a^2*b^2*c* d^2 - 2*a^3*b*d^3)*x^3 + (8*a*b^3*c^3 - 7*a^2*b^2*c^2*d - 2*a^3*b*c*d^2 - 2*a^4*d^3)*x^2)*sqrt(b*d*e)*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*((8*b^4*c^2*d - 3*a *b^3*c*d^2 - 2*a^2*b^2*d^3)*x^3 + (8*a*b^3*c^2*d - 3*a^2*b^2*c*d^2 - 2*a^3 *b*d^3)*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^3*c^3*d - a^4*b^2*c^2*d^2)*e^3 *x^3 + (a^4*b^2*c^3*d - a^5*b*c^2*d^2)*e^3*x^2)
\[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(1/(e*x)**(5/2)/(b*x+a)**(3/2)/(d*x+c)**(1/2),x)
Output:
Integral(1/((e*x)**(5/2)*(a + b*x)**(3/2)*sqrt(c + d*x)), x)
\[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*(e*x)^(5/2)), x)
\[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/((e*x)^(5/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)),x)
Output:
int(1/((e*x)^(5/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {1}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
int(1/(e*x)^(5/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x)
Output:
(sqrt(e)*( - 2*sqrt(c + d*x)*sqrt(a + b*x)*a*c + 4*sqrt(c + d*x)*sqrt(a + b*x)*a*d*x + 8*sqrt(c + d*x)*sqrt(a + b*x)*b*c*x + 2*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x)*x)/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b* c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x) *a**2*b*d**2*x + 4*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x)*x)/(sqrt(x)*a* *2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt( x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a*b**2*c*d*x + 2*sqrt(x)*int((sqr t(c + d*x)*sqrt(a + b*x)*x)/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x) *a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x** 3),x)*a*b**2*d**2*x**2 + 4*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x)*x)/(sq rt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*b**3*c*d*x**2 + sqrt(x)*i nt((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sq rt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2* d*x**3),x)*a**2*b*c*d*x + 8*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqr t(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a*b**2*c**2*x + sqrt(x)*in t((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqr t(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d *x**3),x)*a*b**2*c*d*x**2 + 8*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))...