Integrand size = 26, antiderivative size = 406 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=\frac {2 \sqrt {c+d x}}{a e (e x)^{5/2} \sqrt {a+b x}}-\frac {4 \left (24 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {c+d x}}{15 a^3 c^2 e^3 \sqrt {e x} \sqrt {a+b x}}-\frac {12 \sqrt {a+b x} \sqrt {c+d x}}{5 a^2 e (e x)^{5/2}}+\frac {2 (24 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{15 a^3 c e^2 (e x)^{3/2}}-\frac {4 \sqrt {b} \left (24 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{15 a^{7/2} c^2 e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {b} d (24 b c-a d) \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{15 a^{5/2} c^2 e^{7/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:
2*(d*x+c)^(1/2)/a/e/(e*x)^(5/2)/(b*x+a)^(1/2)-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/e^3/(e*x)^(1/2)/(b*x+a)^(1/2)-12/5*(b*x+a)^ (1/2)*(d*x+c)^(1/2)/a^2/e/(e*x)^(5/2)+2/15*(-a*d+24*b*c)*(b*x+a)^(1/2)*(d* x+c)^(1/2)/a^3/c/e^2/(e*x)^(3/2)-4/15*b^(1/2)*(-a^2*d^2-4*a*b*c*d+24*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^(7/2)/c^2/e^(7/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b *x+a))^(1/2)+2/15*b^(1/2)*d*(-a*d+24*b*c)*(d*x+c)^(1/2)*InverseJacobiAM(ar ctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)),(1-a*d/b/c)^(1/2))/a^(5/2)/c^2/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 = 12.04 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=\frac {x \left (-2 a c (c+d x) \left (-24 b^2 c x^2+a b x (-6 c+d x)+a^2 (3 c+d x)\right )-4 i \sqrt {\frac {a}{b}} b d \left (-24 b^2 c^2+4 a b c d+a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i \sqrt {\frac {a}{b}} b d \left (-24 b^2 c^2+7 a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{15 a^4 c^2 (e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:
Integrate[Sqrt[c + d*x]/((e*x)^(7/2)*(a + b*x)^(3/2)),x]
Output:
(x*(-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*(e*x)^(7/2)*Sqrt[a + b*x]*Sqrt[c + d*x])
Time = 0.57 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.21, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \int -\frac {6 b c-a d+5 b d x}{2 (e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {6 b c-a d+5 b d x}{(e x)^{5/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {e \left (24 b^2 c^2-7 a b d c-2 a^2 d^2+3 b d (6 b c-a d) x\right )}{2 (e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c e^2}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {24 b^2 c^2-7 a b d c-2 a^2 d^2+3 b d (6 b c-a d) x}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {b e \left (c \left (48 b^2 c^2-32 a b d c-a^2 d^2\right )+d \left (24 b^2 c^2-7 a b d c-2 a^2 d^2\right ) x\right )}{2 \sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e^2}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {b \int \frac {c \left (48 b^2 c^2-32 a b d c-a^2 d^2\right )+d \left (24 b^2 c^2-7 a b d c-2 a^2 d^2\right ) x}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {2 \int -\frac {d (b c-a d) e \left (a c (24 b c-a d)+2 \left (24 b^2 c^2-4 a b d c-a^2 d^2\right ) x\right )}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a e (b c-a d)}+\frac {4 \sqrt {e x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a e \sqrt {a+b x}}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {e x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a e \sqrt {a+b x}}-\frac {d \int \frac {a c (24 b c-a d)+2 \left (24 b^2 c^2-4 a b d c-a^2 d^2\right ) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {e x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a e \sqrt {a+b x}}-\frac {d \left (\frac {2 \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {c \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {e x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a e \sqrt {a+b x}}-\frac {d \left (\frac {2 \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {e x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a e \sqrt {a+b x}}-\frac {d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {e x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a e \sqrt {a+b x}}-\frac {d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {-\frac {-\frac {b \left (\frac {4 \sqrt {e x} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right )}{a e \sqrt {a+b x}}-\frac {d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-a^2 d^2-4 a b c d+24 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \left (-a^2 d^2-32 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{a}\right )}{a c e}-\frac {2 \sqrt {c+d x} \left (-2 a^2 d^2-7 a b c d+24 b^2 c^2\right )}{a c e \sqrt {e x} \sqrt {a+b x}}}{3 a c e}-\frac {2 \sqrt {c+d x} (6 b c-a d)}{3 a c e (e x)^{3/2} \sqrt {a+b x}}}{5 a e}-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2} \sqrt {a+b x}}\) |
Input:
Int[Sqrt[c + d*x]/((e*x)^(7/2)*(a + b*x)^(3/2)),x]
Output:
(-2*Sqrt[c + d*x])/(5*a*e*(e*x)^(5/2)*Sqrt[a + b*x]) - ((-2*(6*b*c - a*d)* Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)*Sqrt[a + b*x]) - ((-2*(24*b^2*c^2 - 7* a*b*c*d - 2*a^2*d^2)*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]*Sqrt[a + b*x]) - (b*( (4*(24*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*Sqrt[e*x]*Sqrt[c + d*x])/(a*e*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[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*(48*b^2*c^2 - 32*a*b*c*d - a^2*d^2)*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*c*e))/(5*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]
Leaf count of result is larger than twice the leaf count of optimal. \(734\) vs. \(2(345)=690\).
Time = 4.72 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.81
| 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^{2} x^{3}}-\frac {2 \left (a d -9 b c \right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{15 e^{4} a^{3} 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^{2} d^{2}+8 a b c d -33 b^{2} c^{2}\right )}{15 e^{4} a^{4} 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}}{a^{4} e^{4} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (-\frac {b d \left (a d -9 b c \right )}{15 e^{3} a^{3} c}+\frac {\left (a d +b c \right ) \left (2 a^{2} d^{2}+8 a b c d -33 b^{2} c^{2}\right )}{15 a^{4} c^{2} e^{3}}-\frac {\left (a d e +b c e \right ) \left (2 a^{2} d^{2}+8 a b c d -33 b^{2} c^{2}\right )}{15 e^{4} a^{4} c^{2}}+\frac {\left (a d -b c \right ) b^{2}}{e^{3} a^{4}}+\frac {b^{3} c}{e^{3} 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 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^{2} d^{2}+8 a b c d -33 b^{2} c^{2}\right )}{15 a^{4} c^{2} e^{3}}+\frac {b^{3} d}{e^{3} 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 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}}\) | \(735\) |
| default | \(\frac {\frac {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^{3} c \,d^{3} x^{2}}{15}+\frac {14 \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}}{15}-\frac {16 \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}}{5}-\frac {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 {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3} x^{2}}{15}-\frac {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 {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}}{5}+\frac {112 \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}}{15}-\frac {32 \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}}{5}+\frac {4 a^{2} b \,d^{4} x^{4}}{15}+\frac {16 a \,b^{2} c \,d^{3} x^{4}}{15}-\frac {32 b^{3} c^{2} d^{2} x^{4}}{5}+\frac {4 a^{3} d^{4} x^{3}}{15}+\frac {6 a^{2} b c \,d^{3} x^{3}}{5}-\frac {32 a \,b^{2} c^{2} d^{2} x^{3}}{15}-\frac {32 b^{3} c^{3} d \,x^{3}}{5}+\frac {2 a^{3} c \,d^{3} x^{2}}{15}+\frac {26 a^{2} b \,c^{2} d^{2} x^{2}}{15}-\frac {16 a \,b^{2} c^{3} d \,x^{2}}{5}-\frac {8 a^{3} c^{2} d^{2} x}{15}+\frac {4 a^{2} b \,c^{3} d x}{5}-\frac {2 c^{3} d \,a^{3}}{5}}{x^{2} c^{2} e^{3} \sqrt {e x}\, d \,a^{4} \sqrt {b x +a}\, \sqrt {x d +c}}\) | \(759\) |
Input:
int((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x+a)^(3/2),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^2*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^3-2/15/e^4/a^3/c*( a*d-9*b*c)*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^2+2/15*(b*d*e*x ^2+a*d*e*x+b*c*e*x+a*c*e)/e^4/a^4/c^2*(2*a^2*d^2+8*a*b*c*d-33*b^2*c^2)/(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)*b^2/a^4/e^4 /((x+a/b)*(b*d*e*x^2+b*c*e*x))^(1/2)+2*(-1/15*b*d/e^3*(a*d-9*b*c)/a^3/c+1/ 15*(a*d+b*c)*(2*a^2*d^2+8*a*b*c*d-33*b^2*c^2)/a^4/c^2/e^3-1/15*(a*d*e+b*c* e)/e^4/a^4/c^2*(2*a^2*d^2+8*a*b*c*d-33*b^2*c^2)+(a*d-b*c)*b^2/e^3/a^4+b^3* c/e^3/a^4)*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*(-1/15*d*b*(2*a^2*d^2+8*a*b*c*d-33*b^2*c ^2)/a^4/c^2/e^3+b^3*d/e^3/a^4)*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))))
Time = 0.12 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (3 \, a^{3} b c^{2} d + 2 \, {\left (24 \, b^{4} c^{2} d - 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + {\left (24 \, a b^{3} c^{2} d - 7 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{2} - {\left (6 \, a^{2} b^{2} c^{2} d - a^{3} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} + {\left ({\left (48 \, b^{4} c^{3} - 32 \, a b^{3} c^{2} d - 7 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{4} + {\left (48 \, a b^{3} c^{3} - 32 \, a^{2} b^{2} c^{2} d - 7 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{3}\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 6 \, {\left ({\left (24 \, b^{4} c^{2} d - 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{4} + {\left (24 \, a b^{3} c^{2} d - 4 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3}\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{45 \, {\left (a^{4} b^{2} c^{2} d e^{4} x^{4} + a^{5} b c^{2} d e^{4} x^{3}\right )}} \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x+a)^(3/2),x, algorithm="fricas")
Output:
-2/45*(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 + a)*sqrt(d*x + c)*sqrt(e*x) + ((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*e)*weierstrassPI nverse(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)) + 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*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^4*b^ 2*c^2*d*e^4*x^4 + a^5*b*c^2*d*e^4*x^3)
Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(1/2)/(e*x)**(7/2)/(b*x+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)/((b*x + a)^(3/2)*(e*x)^(7/2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)/((b*x + a)^(3/2)*(e*x)^(7/2)), x)
Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=\int \frac {\sqrt {c+d\,x}}{{\left (e\,x\right )}^{7/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(1/2)/((e*x)^(7/2)*(a + b*x)^(3/2)),x)
Output:
int((c + d*x)^(1/2)/((e*x)^(7/2)*(a + b*x)^(3/2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x)^{3/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a^{2} x^{3}+2 \sqrt {x}\, a b \,x^{4}+\sqrt {x}\, b^{2} x^{5}}d x \right )}{e^{4}} \] Input:
int((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x+a)^(3/2),x)
Output:
(sqrt(e)*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))/e**4