Integrand size = 26, antiderivative size = 385 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {4 \left (24 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {c+d x}}{15 a^3 c^2 \sqrt {a x+b x^2}}+\frac {2 \sqrt {c+d x}}{a x^2 \sqrt {a x+b x^2}}-\frac {12 \sqrt {c+d x} \sqrt {a x+b x^2}}{5 a^2 x^3}+\frac {2 (24 b c-a d) \sqrt {c+d x} \sqrt {a x+b x^2}}{15 a^3 c x^2}-\frac {4 \sqrt {b} \left (24 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {x} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{7/2} c^2 \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}}+\frac {2 \sqrt {b} d (24 b c-a d) \sqrt {x} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 a^{5/2} c^2 \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}} \] Output:
-4/15*(-a^2*d^2-4*a*b*c*d+24*b^2*c^2)*(d*x+c)^(1/2)/a^3/c^2/(b*x^2+a*x)^(1 /2)+2*(d*x+c)^(1/2)/a/x^2/(b*x^2+a*x)^(1/2)-12/5*(d*x+c)^(1/2)*(b*x^2+a*x) ^(1/2)/a^2/x^3+2/15*(-a*d+24*b*c)*(d*x+c)^(1/2)*(b*x^2+a*x)^(1/2)/a^3/c/x^ 2-4/15*b^(1/2)*(-a^2*d^2-4*a*b*c*d+24*b^2*c^2)*x^(1/2)*(d*x+c)^(1/2)*Ellip ticE(b^(1/2)*x^(1/2)/a^(1/2)/(1+b*x/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(7/2)/c^ 2/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x)^(1/2)+2/15*b^(1/2)*d*(-a*d+24*b* c)*x^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x^(1/2)/a^(1/2)),( 1-a*d/b/c)^(1/2))/a^(5/2)/c^2/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x)^(1/2 )
Result contains complex when optimal does not.
Time = 15.08 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=\frac {-2 a c (c+d x) \left (-24 b^2 c x^2+a b x (-6 c+d x)+a^2 (3 c+d x)\right )-4 i \sqrt {\frac {a}{b}} b d \left (-24 b^2 c^2+4 a b c d+a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i \sqrt {\frac {a}{b}} b d \left (-24 b^2 c^2+7 a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{15 a^4 c^2 x^2 \sqrt {x (a+b x)} \sqrt {c+d x}} \] Input:
Integrate[Sqrt[c + d*x]/(x^2*(a*x + b*x^2)^(3/2)),x]
Output:
(-2*a*c*(c + d*x)*(-24*b^2*c*x^2 + a*b*x*(-6*c + d*x) + a^2*(3*c + d*x)) - (4*I)*Sqrt[a/b]*b*d*(-24*b^2*c^2 + 4*a*b*c*d + 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*(-24*b^2*c^2 + 7*a*b*c*d + 2*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^4*c^2*x^2*Sqrt[x*(a + b*x)]*Sqrt[c + d*x])
Time = 1.02 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {1261, 110, 27, 169, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \int \frac {\sqrt {c+d x}}{x^{7/2} (a+b x)^{3/2}}dx}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 \int -\frac {6 b c-a d+5 b d x}{2 x^{5/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {\int \frac {6 b c-a d+5 b d x}{x^{5/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {2 \int \frac {24 b^2 c^2-7 a b d c-2 a^2 d^2+3 b d (6 b c-a d) x}{2 x^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {\int \frac {24 b^2 c^2-7 a b d c-2 a^2 d^2+3 b d (6 b c-a d) x}{x^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {2 \int \frac {b \left (c \left (48 b^2 c^2-32 a b d c-a^2 d^2\right )+d \left (24 b^2 c^2-7 a b d c-2 a^2 d^2\right ) x\right )}{2 \sqrt {x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \int \frac {c \left (48 b^2 c^2-32 a b d c-a^2 d^2\right )+d \left (24 b^2 c^2-7 a b d c-2 a^2 d^2\right ) x}{\sqrt {x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \left (\frac {2 \int -\frac {d (b c-a d) \left (a c (24 b c-a d)+2 \left (24 b^2 c^2-4 a b d c-a^2 d^2\right ) x\right )}{2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {4 \sqrt {x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a \sqrt {a+b x}}\right )}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a \sqrt {a+b x}}-\frac {d \int \frac {a c (24 b c-a d)+2 \left (24 b^2 c^2-4 a b d c-a^2 d^2\right ) x}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a}\right )}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a \sqrt {a+b x}}-\frac {d \left (\frac {2 \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {c \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a \sqrt {a+b x}}-\frac {d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a \sqrt {a+b x}}-\frac {d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a}\right )}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a \sqrt {a+b x}}-\frac {d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a}\right )}{a c}-\frac {2 \sqrt {c+d x} \left (\frac {24 b^2 c}{a}-\frac {2 a d^2}{c}-7 b d\right )}{\sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c x^{3/2} \sqrt {a+b x}}}{5 a}-\frac {2 \sqrt {c+d x}}{5 a x^{5/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[Sqrt[c + d*x]/(x^2*(a*x + b*x^2)^(3/2)),x]
Output:
(x^(3/2)*(a + b*x)^(3/2)*((-2*Sqrt[c + d*x])/(5*a*x^(5/2)*Sqrt[a + b*x]) - ((-2*(6*b*c - a*d)*Sqrt[c + d*x])/(3*a*c*x^(3/2)*Sqrt[a + b*x]) - ((-2*(( 24*b^2*c)/a - 7*b*d - (2*a*d^2)/c)*Sqrt[c + d*x])/(Sqrt[x]*Sqrt[a + b*x]) - (b*((4*(24*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*Sqrt[x]*Sqrt[c + d*x])/(a*Sqrt [a + b*x]) - (d*((4*Sqrt[-a]*(24*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*Sqrt[1 + ( b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/ (b*c)])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (2*Sqrt[-a]*c*(48*b^ 2*c^2 - 32*a*b*c*d - a^2*d^2)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*Elliptic F[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[a + b* x]*Sqrt[c + d*x])))/a))/(a*c))/(3*a*c))/(5*a)))/(a*x + b*x^2)^(3/2)
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]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 3.08 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.54
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (b x +a \right ) \left (d x +c \right )}\, \left (-\frac {2 \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}{5 a^{2} x^{3}}-\frac {2 \left (a d -9 b c \right ) \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}{15 a^{3} c \,x^{2}}+\frac {2 \left (b d \,x^{2}+a d x +c b x +a c \right ) \left (2 a^{2} d^{2}+8 a b c d -33 b^{2} c^{2}\right )}{15 a^{4} c^{2} \sqrt {x \left (b d \,x^{2}+a d x +c b x +a c \right )}}-\frac {2 \left (b d \,x^{2}+c b x \right ) b^{2}}{a^{4} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d \,x^{2}+c b x \right )}}+\frac {2 \left (-\frac {b d \left (a d -9 b c \right )}{15 c \,a^{3}}+\frac {b^{2} \left (a d -b c \right )}{a^{4}}+\frac {b^{3} c}{a^{4}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}+\frac {2 \left (-\frac {b d \left (2 a^{2} d^{2}+8 a b c d -33 b^{2} c^{2}\right )}{15 a^{4} c^{2}}+\frac {b^{3} d}{a^{4}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}\) | \(592\) |
| default | \(\frac {2 \left (2 x^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}+7 x^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}-24 x^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d -2 x^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}-6 x^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}+56 x^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d -48 x^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{3} c^{4}+2 x^{4} a^{2} b \,d^{4}+8 a \,b^{2} c \,d^{3} x^{4}-48 b^{3} c^{2} d^{2} x^{4}+2 a^{3} d^{4} x^{3}+9 x^{3} a^{2} b c \,d^{3}-16 b^{2} c^{2} d^{2} a \,x^{3}-48 b^{3} c^{3} d \,x^{3}+x^{2} a^{3} c \,d^{3}+13 x^{2} a^{2} b \,c^{2} d^{2}-24 a \,b^{2} c^{3} d \,x^{2}-4 x \,a^{3} c^{2} d^{2}+6 x \,a^{2} b \,c^{3} d -3 a^{3} c^{3} d \right ) \sqrt {x \left (b x +a \right )}}{15 x^{3} \left (b x +a \right ) c^{2} d \,a^{4} \sqrt {d x +c}}\) | \(760\) |
Input:
int((d*x+c)^(1/2)/x^2/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
(x*(b*x+a)*(d*x+c))^(1/2)/(x*(b*x+a))^(1/2)/(d*x+c)^(1/2)*(-2/5/a^2/x^3*(b *d*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)-2/15/a^3/c*(a*d-9*b*c)*(b*d*x^3+a*d*x^ 2+b*c*x^2+a*c*x)^(1/2)/x^2+2/15*(b*d*x^2+a*d*x+b*c*x+a*c)/a^4/c^2*(2*a^2*d ^2+8*a*b*c*d-33*b^2*c^2)/(x*(b*d*x^2+a*d*x+b*c*x+a*c))^(1/2)-2*(b*d*x^2+b* c*x)*b^2/a^4/((x+a/b)*(b*d*x^2+b*c*x))^(1/2)+2*(-1/15*b*d*(a*d-9*b*c)/c/a^ 3+b^2*(a*d-b*c)/a^4+1/a^4*b^3*c)*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*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)*EllipticF (((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2*(-1/15*b*d*(2*a^2*d^2+8*a* b*c*d-33*b^2*c^2)/a^4/c^2+1/a^4*b^3*d)*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*x^3+a*d*x^2+b*c*x^2+a*c*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))))
Time = 0.10 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (48 \, b^{4} c^{3} - 32 \, a b^{3} c^{2} d - 7 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{4} + {\left (48 \, a b^{3} c^{3} - 32 \, a^{2} b^{2} c^{2} d - 7 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{3}\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 6 \, {\left ({\left (24 \, b^{4} c^{2} d - 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{4} + {\left (24 \, a b^{3} c^{2} d - 4 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3}\right )} \sqrt {b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + 3 \, {\left (3 \, a^{3} b c^{2} d + 2 \, {\left (24 \, b^{4} c^{2} d - 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + {\left (24 \, a b^{3} c^{2} d - 7 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{2} - {\left (6 \, a^{2} b^{2} c^{2} d - a^{3} b c d^{2}\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {d x + c}\right )}}{45 \, {\left (a^{4} b^{2} c^{2} d x^{4} + a^{5} b c^{2} d x^{3}\right )}} \] Input:
integrate((d*x+c)^(1/2)/x^2/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
-2/45*(((48*b^4*c^3 - 32*a*b^3*c^2*d - 7*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x^4 + (48*a*b^3*c^3 - 32*a^2*b^2*c^2*d - 7*a^3*b*c*d^2 - 2*a^4*d^3)*x^3)*sqrt( b*d)*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)) + 6*((24*b^4*c^2*d - 4*a*b^3*c*d^2 - a^2*b^2*d ^3)*x^4 + (24*a*b^3*c^2*d - 4*a^2*b^2*c*d^2 - a^3*b*d^3)*x^3)*sqrt(b*d)*we ierstrassZeta(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), weierstrassPInve rse(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^3*b*c^2*d + 2*(24*b^4*c^2*d - 4*a*b^3*c*d^2 - a^2*b^2*d^3 )*x^3 + (24*a*b^3*c^2*d - 7*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x^2 - (6*a^2*b^2* c^2*d - a^3*b*c*d^2)*x)*sqrt(b*x^2 + a*x)*sqrt(d*x + c))/(a^4*b^2*c^2*d*x^ 4 + a^5*b*c^2*d*x^3)
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x}}{x^{2} \left (x \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(1/2)/x**2/(b*x**2+a*x)**(3/2),x)
Output:
Integral(sqrt(c + d*x)/(x**2*(x*(a + b*x))**(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x^2/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)/((b*x^2 + a*x)^(3/2)*x^2), x)
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/x^2/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)/((b*x^2 + a*x)^(3/2)*x^2), x)
Timed out. \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c+d\,x}}{x^2\,{\left (b\,x^2+a\,x\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(1/2)/(x^2*(a*x + b*x^2)^(3/2)),x)
Output:
int((c + d*x)^(1/2)/(x^2*(a*x + b*x^2)^(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{x^2 \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a^{2} x^{3}+2 \sqrt {x}\, a b \,x^{4}+\sqrt {x}\, b^{2} x^{5}}d x \] Input:
int((d*x+c)^(1/2)/x^2/(b*x^2+a*x)^(3/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*x**3 + 2*sqrt(x)*a*b*x**4 + sqrt(x)*b**2*x**5),x)