Integrand size = 28, antiderivative size = 369 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {c} (2 A e (2 c d-b e)-B d (c d+b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 d^2 e (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (B d-A e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 d e (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
2/3*(2*A*e*(-b*e+2*c*d)-B*d*(b*e+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^2/e/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*(-A*e+B*d)*Elli pticF(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)*(1+e*x/d)^(1/2)/d/e/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x )^(1/2)+2/3*(-A*e+B*d)*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^(3/2)-2/3*(2 *A*e*(-b*e+2*c*d)-B*d*(b*e+c*d))*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d )^(1/2)
Result contains complex when optimal does not.
Time = 23.43 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \left (b e x (b+c x) \left (B d \left (b e^2 x+c d (2 d+e x)\right )+A e (b e (3 d+2 e x)-c d (5 d+4 e x))\right )-\sqrt {\frac {b}{c}} c (d+e x) \left (\sqrt {\frac {b}{c}} (2 A e (-2 c d+b e)+B d (c d+b e)) (b+c x) (d+e x)+i b e (2 A e (-2 c d+b e)+B d (c d+b e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i e (c d-b e) (3 A c d-b (B d+2 A e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b d^2 e (c d-b e)^2 \sqrt {x (b+c x)} (d+e x)^{3/2}} \]
(2*(b*e*x*(b + c*x)*(B*d*(b*e^2*x + c*d*(2*d + e*x)) + A*e*(b*e*(3*d + 2*e *x) - c*d*(5*d + 4*e*x))) - Sqrt[b/c]*c*(d + e*x)*(Sqrt[b/c]*(2*A*e*(-2*c* d + b*e) + B*d*(c*d + b*e))*(b + c*x)*(d + e*x) + I*b*e*(2*A*e*(-2*c*d + b *e) + B*d*(c*d + b*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellipti cE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*e*(c*d - b*e)*(3*A*c*d - b*(B*d + 2*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I* ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b*d^2*e*(c*d - b*e)^2*Sqrt[ x*(b + c*x)]*(d + e*x)^(3/2))
Time = 0.67 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1237, 27, 1237, 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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {b x+c x^2} (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {2 \int \frac {b B d-3 A c d+2 A b e-c (B d-A e) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\int \frac {b B d-3 A c d+2 A b e-c (B d-A e) x}{(d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}-\frac {2 \int -\frac {c (d (2 b B d-3 A c d+A b e)-(2 A e (2 c d-b e)-B d (c d+b e)) x)}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \int \frac {d (2 b B d-3 A c d+A b e)-(2 A e (2 c d-b e)-B d (c d+b e)) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {d (B d-A e) (c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}-\frac {(2 A e (2 c d-b e)-B d (b e+c d)) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {d \sqrt {x} \sqrt {b+c x} (B d-A e) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x} (2 A e (2 c d-b e)-B d (b e+c d)) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {d \sqrt {x} \sqrt {b+c x} (B d-A e) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {d \sqrt {x} \sqrt {b+c x} (B d-A e) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d-A e) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d-A e) (c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}\) |
(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - ((2* (2*A*e*(2*c*d - b*e) - B*d*(c*d + b*e))*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)* Sqrt[d + e*x]) + (c*((-2*Sqrt[-b]*(2*A*e*(2*c*d - b*e) - B*d*(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)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(B*d - A*e)*(c*d - b*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)])/(Sqr t[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])))/(d*(c*d - b*e)))/(3*d*(c*d - b*e ))
3.13.69.3.1 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[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Time = 1.19 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.63
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 \left (A e -B d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{2} d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (2 A b \,e^{2}-4 A c d e +B b d e +B c \,d^{2}\right )}{3 e \,d^{2} \left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c \left (A e -B d \right )}{3 e d \left (b e -c d \right )}+\frac {2 A b \,e^{2}-4 A c d e +B b d e +B c \,d^{2}}{3 \left (b e -c d \right ) e \,d^{2}}-\frac {b \left (2 A b \,e^{2}-4 A c d e +B b d e +B c \,d^{2}\right )}{3 d^{2} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}-\frac {2 \left (2 A b \,e^{2}-4 A c d e +B b d e +B c \,d^{2}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{3 d^{2} \left (b e -c d \right )^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(602\) |
| default | \(\text {Expression too large to display}\) | \(1635\) |
((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/3/e^2/d/(b*e- c*d)*(A*e-B*d)*(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)/e/d^2/(b*e-c*d)^2*(2*A*b*e^2-4*A*c*d*e+B*b*d*e+B*c*d^2)/((x+d/e)* (c*e*x^2+b*e*x))^(1/2)+2*(1/3*c/e*(A*e-B*d)/d/(b*e-c*d)+1/3/(b*e-c*d)/e*(2 *A*b*e^2-4*A*c*d*e+B*b*d*e+B*c*d^2)/d^2-1/3*b/d^2/(b*e-c*d)^2*(2*A*b*e^2-4 *A*c*d*e+B*b*d*e+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)*EllipticF(((x+b/c )/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))-2/3*(2*A*b*e^2-4*A*c*d*e+B*b*d*e+B*c *d^2)/d^2/(b*e-c*d)^2*b*((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 = 0.11 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.12 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (B c^{2} d^{5} + 2 \, A b^{2} d^{2} e^{3} - {\left (4 \, B b c - 5 \, A c^{2}\right )} d^{4} e + {\left (B b^{2} - 5 \, A b c\right )} d^{3} e^{2} + {\left (B c^{2} d^{3} e^{2} + 2 \, A b^{2} e^{5} - {\left (4 \, B b c - 5 \, A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} - 5 \, A b c\right )} d e^{4}\right )} x^{2} + 2 \, {\left (B c^{2} d^{4} e + 2 \, A b^{2} d e^{4} - {\left (4 \, B b c - 5 \, A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} - 5 \, 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 (B c^{2} d^{4} e + 2 \, A b c d^{2} e^{3} + {\left (B b c - 4 \, A c^{2}\right )} d^{3} e^{2} + {\left (B c^{2} d^{2} e^{3} + 2 \, A b c e^{5} + {\left (B b c - 4 \, A c^{2}\right )} d e^{4}\right )} x^{2} + 2 \, {\left (B c^{2} d^{3} e^{2} + 2 \, A b c d e^{4} + {\left (B b c - 4 \, 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 (2 \, B c^{2} d^{3} e^{2} - 5 \, A c^{2} d^{2} e^{3} + 3 \, A b c d e^{4} + {\left (B c^{2} d^{2} e^{3} + 2 \, A b c e^{5} + {\left (B b c - 4 \, A c^{2}\right )} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (c^{3} d^{6} e^{2} - 2 \, b c^{2} d^{5} e^{3} + b^{2} c d^{4} e^{4} + {\left (c^{3} d^{4} e^{4} - 2 \, b c^{2} d^{3} e^{5} + b^{2} c d^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{3} d^{5} e^{3} - 2 \, b c^{2} d^{4} e^{4} + b^{2} c d^{3} e^{5}\right )} x\right )}} \]
2/9*((B*c^2*d^5 + 2*A*b^2*d^2*e^3 - (4*B*b*c - 5*A*c^2)*d^4*e + (B*b^2 - 5 *A*b*c)*d^3*e^2 + (B*c^2*d^3*e^2 + 2*A*b^2*e^5 - (4*B*b*c - 5*A*c^2)*d^2*e ^3 + (B*b^2 - 5*A*b*c)*d*e^4)*x^2 + 2*(B*c^2*d^4*e + 2*A*b^2*d*e^4 - (4*B* b*c - 5*A*c^2)*d^3*e^2 + (B*b^2 - 5*A*b*c)*d^2*e^3)*x)*sqrt(c*e)*weierstra ssPInverse(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*(B*c^2*d^4*e + 2*A*b*c*d^2*e^3 + (B*b*c - 4*A*c^2)*d^3*e ^2 + (B*c^2*d^2*e^3 + 2*A*b*c*e^5 + (B*b*c - 4*A*c^2)*d*e^4)*x^2 + 2*(B*c^ 2*d^3*e^2 + 2*A*b*c*d*e^4 + (B*b*c - 4*A*c^2)*d^2*e^3)*x)*sqrt(c*e)*weiers trassZeta(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), weierstrassPInverse( 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*(2*B*c^2*d^3*e^2 - 5*A*c^2*d^2*e^3 + 3*A*b*c*d*e^4 + (B*c^2*d^2*e^ 3 + 2*A*b*c*e^5 + (B*b*c - 4*A*c^2)*d*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^3*d^6*e^2 - 2*b*c^2*d^5*e^3 + b^2*c*d^4*e^4 + (c^3*d^4*e^4 - 2*b*c ^2*d^3*e^5 + b^2*c*d^2*e^6)*x^2 + 2*(c^3*d^5*e^3 - 2*b*c^2*d^4*e^4 + b^2*c *d^3*e^5)*x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]