Integrand size = 26, antiderivative size = 372 \[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {c+d x}}{15 a^2 c^3 e^3 \sqrt {e x} \sqrt {a+b x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}+\frac {8 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{15 a^2 c^2 e^2 (e x)^{3/2}}-\frac {2 \sqrt {b} \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} c^3 e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {8 \sqrt {b} d (b c+a d) \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{15 a^{3/2} c^3 e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:
-2/15*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^2)*(d*x+c)^(1/2)/a^2/c^3/e^3/(e*x)^(1/2 )/(b*x+a)^(1/2)-2/5*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c/e/(e*x)^(5/2)+8/15*(a* d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2/e^2/(e*x)^(3/2)-2/15*b^(1/2)*(8 *a^2*d^2+7*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^3/e^(7/2)/(b* x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)+8/15*b^(1/2)*d*(a*d+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^3/e^(7/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)
Result contains complex when optimal does not.
Time = 5.22 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {x \left (2 a c (a+b x) (c+d x) (-3 a c+4 b c x+4 a d x)+2 i \sqrt {\frac {a}{b}} b d \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} b d \left (4 b^2 c^2+3 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{15 a^3 c^3 (e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:
Integrate[1/((e*x)^(7/2)*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
Output:
(x*(2*a*c*(a + b*x)*(c + d*x)*(-3*a*c + 4*b*c*x + 4*a*d*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^(7/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] - (2*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^(7/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)])) /(15*a^3*c^3*(e*x)^(7/2)*Sqrt[a + b*x]*Sqrt[c + d*x])
Time = 0.47 (sec) , antiderivative size = 431, 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 = {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.
| \(\displaystyle \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle -\frac {2 \int \frac {e (4 (b c+a d)+3 b d x)}{2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{5 a c e^2}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 (b c+a d)+3 b d x}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {e \left (8 b^2 c^2+7 a b d c+8 a^2 d^2+4 b d (b c+a d) x\right )}{2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c e^2}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {8 b^2 c^2+7 a b d c+8 a^2 d^2+4 b d (b c+a d) x}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {2 \int -\frac {b d e \left (4 a c (b c+a d)+\left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) x\right )}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c e^2}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {b d \int \frac {4 a c (b c+a d)+\left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {-\frac {\frac {b d \left (\frac {\left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {c \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle -\frac {-\frac {\frac {b d \left (\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle -\frac {-\frac {\frac {b d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {-\frac {\frac {b d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {-\frac {\frac {b d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (4 a^2 d^2+3 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {8 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\) |
Input:
Int[1/((e*x)^(7/2)*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
Output:
(-2*Sqrt[a + b*x]*Sqrt[c + d*x])/(5*a*c*e*(e*x)^(5/2)) - ((-8*(b*c + a*d)* Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)) - ((-2*(8*b^2*c^2 + 7*a *b*c*d + 8*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]) + (b*d* ((2*Sqrt[-a]*(8*b^2*c^2 + 7*a*b*c*d + 8*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*( 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]*Ell ipticF[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*c*e))/(3*a*c*e))/(5*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. \(640\) vs. \(2(317)=634\).
Time = 4.54 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.72
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{5 e^{4} a c \,x^{3}}+\frac {8 \left (a d +b c \right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{15 a^{2} c^{2} e^{4} x^{2}}-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right ) \left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right )}{15 e^{4} a^{3} c^{3} \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (\frac {4 d b \left (a d +b c \right )}{15 a^{2} c^{2} e^{3}}-\frac {\left (a d +b c \right ) \left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right )}{15 a^{3} c^{3} e^{3}}+\frac {\left (a d e +b c e \right ) \left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right )}{15 e^{4} a^{3} c^{3}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 b \left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{15 a^{3} c^{2} e^{3} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\) | \(641\) |
| 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} x^{2}+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} x^{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 \,x^{2}-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} x^{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^{2} b \,c^{2} d^{2} x^{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 \,x^{2}+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} x^{2}+8 a^{2} b \,d^{4} x^{4}+7 a \,b^{2} c \,d^{3} x^{4}+8 b^{3} c^{2} d^{2} x^{4}+8 a^{3} d^{4} x^{3}+11 a^{2} b c \,d^{3} x^{3}+11 a \,b^{2} c^{2} d^{2} x^{3}+8 b^{3} c^{3} d \,x^{3}+4 a^{3} c \,d^{3} x^{2}+2 a^{2} b \,c^{2} d^{2} x^{2}+4 a \,b^{2} c^{3} d \,x^{2}-a^{3} c^{2} d^{2} x -a^{2} b \,c^{3} d x +3 c^{3} d \,a^{3}\right ) \sqrt {x d +c}\, \sqrt {b x +a}}{15 x^{2} c^{3} a^{3} d \left (b d \,x^{2}+a d x +b c x +a c \right ) e^{3} \sqrt {e x}}\) | \(779\) |
Input:
int(1/(e*x)^(7/2)/(b*x+a)^(1/2)/(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)*(-2/5/ e^4/a/c*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^3+8/15*(a*d+b*c)/a ^2/c^2/e^4*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^2-2/15*(b*d*e*x ^2+a*d*e*x+b*c*e*x+a*c*e)/e^4/a^3/c^3*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^2)/(x*( b*d*e*x^2+a*d*e*x+b*c*e*x+a*c*e))^(1/2)+2*(4/15*d*b*(a*d+b*c)/a^2/c^2/e^3- 1/15*(a*d+b*c)*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^2)/a^3/c^3/e^3+1/15*(a*d*e+b*c *e)/e^4/a^3/c^3*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^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)*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2/15 *b*(8*a^2*d^2+7*a*b*c*d+8*b^2*c^2)/a^3/c^2/e^3*((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 = 0.08 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left ({\left (8 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3}\right )} \sqrt {b d e} x^{3} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (8 \, b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} \sqrt {b d e} x^{3} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + 3 \, {\left (3 \, a^{2} b c^{2} d + {\left (8 \, b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} x^{2} - 4 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}\right )}}{45 \, a^{3} b c^{3} d e^{4} x^{3}} \] Input:
integrate(1/(e*x)^(7/2)/(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) *x^3*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)*x^3*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*a^2*b*c^2*d + (8*b^3*c^2*d + 7*a*b^2*c* d^2 + 8*a^2*b*d^3)*x^2 - 4*(a*b^2*c^2*d + a^2*b*c*d^2)*x)*sqrt(b*x + a)*sq rt(d*x + c)*sqrt(e*x))/(a^3*b*c^3*d*e^4*x^3)
Timed out. \[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)**(7/2)/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
Output:
Timed out
\[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(7/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*(e*x)^(7/2)), x)
\[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(7/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*(e*x)^(7/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{7/2}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/((e*x)^(7/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int(1/((e*x)^(7/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {1}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {d x +c}\, \sqrt {b x +a}\, a c +6 \sqrt {d x +c}\, \sqrt {b x +a}\, b d \,x^{2}-4 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d \,x^{5}+a d \,x^{4}+b c \,x^{4}+a c \,x^{3}}d x \right ) a^{2} c d \,x^{2}-4 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d \,x^{5}+a d \,x^{4}+b c \,x^{4}+a c \,x^{3}}d x \right ) a b \,c^{2} x^{2}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d \,x^{2}+a d x +b c x +a c}d x \right ) b^{2} d^{2} x^{2}\right )}{5 \sqrt {x}\, a^{2} c^{2} e^{4} x^{2}} \] Input:
int(1/(e*x)^(7/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x)
Output:
(sqrt(e)*( - 2*sqrt(c + d*x)*sqrt(a + b*x)*a*c + 6*sqrt(c + d*x)*sqrt(a + b*x)*b*d*x**2 - 4*sqrt(x)*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a*c*x **3 + a*d*x**4 + b*c*x**4 + b*d*x**5),x)*a**2*c*d*x**2 - 4*sqrt(x)*int((sq rt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a*c*x**3 + a*d*x**4 + b*c*x**4 + b*d*x **5),x)*a*b*c**2*x**2 - 3*sqrt(x)*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x) )/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**2*d**2*x**2))/(5*sqrt(x)*a**2*c** 2*e**4*x**2)