Integrand size = 28, antiderivative size = 346 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (d^2 (4 B c d-3 b B e-A c e)+e (B d (5 c d-4 b e)-A e (2 c d-b e)) x\right ) \sqrt {b x+c x^2}}{3 d e^2 (c d-b e) (d+e x)^{3/2}}+\frac {2 \sqrt {-b} \sqrt {c} (B d (8 c d-7 b e)-A e (2 c d-b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 d e^3 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} (8 B c d-3 b B e-2 A c e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
2/3*(B*d*(-7*b*e+8*c*d)-A*e*(-b*e+2*c*d))*EllipticE(c^(1/2)*x^(1/2)/(-b)^( 1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^( 1/2)/d/e^3/(-b*e+c*d)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/3*(-2*A*c*e-3*B* b*e+8*B*c*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1 /2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/e^3/c^(1/2)/(e*x+d)^(1/2)/(c*x ^2+b*x)^(1/2)-2/3*(d^2*(-A*c*e-3*B*b*e+4*B*c*d)+e*(B*d*(-4*b*e+5*c*d)-A*e* (-b*e+2*c*d))*x)*(c*x^2+b*x)^(1/2)/d/e^2/(-b*e+c*d)/(e*x+d)^(3/2)
Result contains complex when optimal does not.
Time = 16.64 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 \left (\sqrt {\frac {b}{c}} e x (b+c x) \left (A e \left (-b e^2 x+c d (d+2 e x)\right )+B d (b e (3 d+4 e x)-c d (4 d+5 e x))\right )+(d+e x) \left (\sqrt {\frac {b}{c}} (B d (8 c d-7 b e)+A e (-2 c d+b e)) (b+c x) (d+e x)-i b e (A e (2 c d-b e)+B d (-8 c d+7 b e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (4 B d-A e) (c d-b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 \sqrt {\frac {b}{c}} d e^3 (c d-b e) \sqrt {x (b+c x)} (d+e x)^{3/2}} \]
(2*(Sqrt[b/c]*e*x*(b + c*x)*(A*e*(-(b*e^2*x) + c*d*(d + 2*e*x)) + B*d*(b*e *(3*d + 4*e*x) - c*d*(4*d + 5*e*x))) + (d + e*x)*(Sqrt[b/c]*(B*d*(8*c*d - 7*b*e) + A*e*(-2*c*d + b*e))*(b + c*x)*(d + e*x) - I*b*e*(A*e*(2*c*d - b*e ) + B*d*(-8*c*d + 7*b*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elli pticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(4*B*d - A*e)*(c* d - b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[S qrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*Sqrt[b/c]*d*e^3*(c*d - b*e)*Sqrt[x*( b + c*x)]*(d + e*x)^(3/2))
Time = 0.60 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1229, 27, 1269, 1169, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {2 \int -\frac {b d (4 B c d-3 b B e-A c e)+c (B d (8 c d-7 b e)-A e (2 c d-b e)) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b d (4 B c d-3 b B e-A c e)+c (B d (8 c d-7 b e)-A e (2 c d-b e)) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {c (B d (8 c d-7 b e)-A e (2 c d-b e)) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (-2 A c e-3 b B e+8 B c d) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {\frac {c \sqrt {x} \sqrt {b+c x} (B d (8 c d-7 b e)-A e (2 c d-b e)) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (-2 A c e-3 b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {\frac {c \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (-2 A c e-3 b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (-2 A c e-3 b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (-2 A c e-3 b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (-2 A c e-3 b B e+8 B c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\) |
(-2*(d^2*(4*B*c*d - 3*b*B*e - A*c*e) + e*(B*d*(5*c*d - 4*b*e) - A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(3*d*e^2*(c*d - b*e)*(d + e*x)^(3/2)) + ((2 *Sqrt[-b]*Sqrt[c]*(B*d*(8*c*d - 7*b*e) - A*e*(2*c*d - b*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b *e)/(c*d)])/(e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(8*B*c*d - 3*b*B*e - 2*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e* x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c] *e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]))/(3*d*e^2*(c*d - b*e))
3.13.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(616\) vs. \(2(298)=596\).
Time = 1.30 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (A e -B d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{4} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (A b \,e^{2}-2 A c d e -4 B b d e +5 B c \,d^{2}\right )}{3 d \left (b e -c d \right ) e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {A c e +B b e -2 B c d}{e^{3}}-\frac {\left (A e -B d \right ) c}{3 e^{3}}+\frac {A b \,e^{2}-2 A c d e -4 B b d e +5 B c \,d^{2}}{3 e^{3} d}-\frac {b \left (A b \,e^{2}-2 A c d e -4 B b d e +5 B c \,d^{2}\right )}{3 e^{2} d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {B c}{e^{2}}-\frac {c \left (A b \,e^{2}-2 A c d e -4 B b d e +5 B c \,d^{2}\right )}{3 e^{2} d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\) | \(617\) |
| default | \(\text {Expression too large to display}\) | \(1959\) |
1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((e*x+d)*x*(c*x+b))^(1/2)/x/(c*x+b)*(-2/ 3*(A*e-B*d)/e^4*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^2+2/3*(c*e*x ^2+b*e*x)/d/(b*e-c*d)/e^3*(A*b*e^2-2*A*c*d*e-4*B*b*d*e+5*B*c*d^2)/((x+d/e) *(c*e*x^2+b*e*x))^(1/2)+2*((A*c*e+B*b*e-2*B*c*d)/e^3-1/3*(A*e-B*d)/e^3*c+1 /3/e^3*(A*b*e^2-2*A*c*d*e-4*B*b*d*e+5*B*c*d^2)/d-1/3*b/e^2/d/(b*e-c*d)*(A* b*e^2-2*A*c*d*e-4*B*b*d*e+5*B*c*d^2))*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b /c+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*Ellipt icF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))+2*(B*c/e^2-1/3*c/e^2*(A*b *e^2-2*A*c*d*e-4*B*b*d*e+5*B*c*d^2)/d/(b*e-c*d))*b/c*((x+b/c)/b*c)^(1/2)*( (x+d/e)/(-b/c+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^( 1/2)*((-b/c+d/e)*EllipticE(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))-d/ e*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 748, normalized size of antiderivative = 2.16 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} - {\left (11 \, B b c + 2 \, A c^{2}\right )} d^{4} e + 2 \, {\left (B b^{2} + A b c\right )} d^{3} e^{2} + {\left (8 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - {\left (11 \, B b c + 2 \, A c^{2}\right )} d^{2} e^{3} + 2 \, {\left (B b^{2} + A b c\right )} d e^{4}\right )} x^{2} + 2 \, {\left (8 \, B c^{2} d^{4} e + A b^{2} d e^{4} - {\left (11 \, B b c + 2 \, A c^{2}\right )} d^{3} e^{2} + 2 \, {\left (B b^{2} + A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (8 \, B c^{2} d^{4} e + A b c d^{2} e^{3} - {\left (7 \, B b c + 2 \, A c^{2}\right )} d^{3} e^{2} + {\left (8 \, B c^{2} d^{2} e^{3} + A b c e^{5} - {\left (7 \, B b c + 2 \, A c^{2}\right )} d e^{4}\right )} x^{2} + 2 \, {\left (8 \, B c^{2} d^{3} e^{2} + A b c d e^{4} - {\left (7 \, B b c + 2 \, A c^{2}\right )} d^{2} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (4 \, B c^{2} d^{3} e^{2} - {\left (3 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (5 \, B c^{2} d^{2} e^{3} + A b c e^{5} - 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (c^{2} d^{4} e^{4} - b c d^{3} e^{5} + {\left (c^{2} d^{2} e^{6} - b c d e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{3} e^{5} - b c d^{2} e^{6}\right )} x\right )}} \]
-2/9*((8*B*c^2*d^5 + A*b^2*d^2*e^3 - (11*B*b*c + 2*A*c^2)*d^4*e + 2*(B*b^2 + A*b*c)*d^3*e^2 + (8*B*c^2*d^3*e^2 + A*b^2*e^5 - (11*B*b*c + 2*A*c^2)*d^ 2*e^3 + 2*(B*b^2 + A*b*c)*d*e^4)*x^2 + 2*(8*B*c^2*d^4*e + A*b^2*d*e^4 - (1 1*B*b*c + 2*A*c^2)*d^3*e^2 + 2*(B*b^2 + A*b*c)*d^2*e^3)*x)*sqrt(c*e)*weier strassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d ^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(8*B*c^2*d^4*e + A*b*c*d^2*e^3 - (7*B*b*c + 2*A*c^2) *d^3*e^2 + (8*B*c^2*d^2*e^3 + A*b*c*e^5 - (7*B*b*c + 2*A*c^2)*d*e^4)*x^2 + 2*(8*B*c^2*d^3*e^2 + A*b*c*d*e^4 - (7*B*b*c + 2*A*c^2)*d^2*e^3)*x)*sqrt(c *e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2* c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstras sPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(4*B*c^2*d^3*e^2 - (3*B*b*c + A*c^2)*d^2*e^3 + (5*B*c^2* d^2*e^3 + A*b*c*e^5 - 2*(2*B*b*c + A*c^2)*d*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt (e*x + d))/(c^2*d^4*e^4 - b*c*d^3*e^5 + (c^2*d^2*e^6 - b*c*d*e^7)*x^2 + 2* (c^2*d^3*e^5 - b*c*d^2*e^6)*x)
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]