Integrand size = 23, antiderivative size = 309 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (8 c d-7 b e-6 c e x) \sqrt {b x+c x^2}}{5 e^3}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {2 \sqrt {-b} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 \sqrt {c} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {16 \sqrt {-b} d (c d-b e) (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{5 \sqrt {c} e^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2*(c*x^2+b*x)^(3/2)/e/(e*x+d)^(1/2)+2/5*(b^2*e^2-16*b*c*d*e+16*c^2*d^2)*E llipticE(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)*(e*x+d)^(1/2)/e^4/c^(1/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)- 16/5*d*(-b*e+c*d)*(-b*e+2*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^4/c^(1/2)/ (e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/5*(-6*c*e*x-7*b*e+8*c*d)*(e*x+d)^(1/2)*( c*x^2+b*x)^(1/2)/e^3
Result contains complex when optimal does not.
Time = 21.93 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.99 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \left ((b+c x) \left (b^2 e^2 (d+e x)+b c e \left (-16 d^2-9 d e x+2 e^2 x^2\right )+c^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+i \sqrt {\frac {b}{c}} c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i \sqrt {\frac {b}{c}} c e \left (8 c^2 d^2-9 b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{5 c e^4 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
(2*((b + c*x)*(b^2*e^2*(d + e*x) + b*c*e*(-16*d^2 - 9*d*e*x + 2*e^2*x^2) + c^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) + I*Sqrt[b/c]*c*e*(16*c ^2*d^2 - 16*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2) *EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(8 *c^2*d^2 - 9*b*c*d*e + b^2*e^2)*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)]))/(5*c*e^4*Sqrt[x*( b + c*x)]*Sqrt[d + e*x])
Time = 0.50 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1161, 1231, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {3 \int \frac {(b+2 c x) \sqrt {c x^2+b x}}{\sqrt {d+e x}}dx}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {3 \left (-\frac {2 \int -\frac {c \left (b d (8 c d-7 b e)+\left (16 c^2 d^2-16 b c e d+b^2 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {b d (8 c d-7 b e)+\left (16 c^2 d^2-16 b c e d+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {8 d (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\sqrt {x} \sqrt {b+c x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {8 d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {8 d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {3 \left (\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {8 d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {3 \left (\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {8 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {3 \left (\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {16 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
(-2*(b*x + c*x^2)^(3/2))/(e*Sqrt[d + e*x]) + (3*((-2*Sqrt[d + e*x]*(8*c*d - 7*b*e - 6*c*e*x)*Sqrt[b*x + c*x^2])/(15*e^2) + ((2*Sqrt[-b]*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*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]) - (16*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*Sqrt[x] *Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sq rt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]))/(15*e^ 2)))/e
3.4.95.3.1 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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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]
Leaf count of result is larger than twice the leaf count of optimal. \(662\) vs. \(2(257)=514\).
Time = 1.93 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.15
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 \left (c e \,x^{2}+b e x \right ) d \left (b e -c d \right )}{e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 c x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{2}}+\frac {2 \left (\frac {c \left (2 b e -c d \right )}{e^{2}}-\frac {2 c \left (2 b e +2 c d \right )}{5 e^{2}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (-\frac {d \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}{e^{4}}+\frac {d \left (b e -c d \right )^{2}}{e^{4}}-\frac {b d \left (b e -c d \right )}{e^{3}}-\frac {\left (\frac {c \left (2 b e -c d \right )}{e^{2}}-\frac {2 c \left (2 b e +2 c d \right )}{5 e^{2}}\right ) b d}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \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 (\frac {b}{c}+x \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^{2} e^{2}-2 b c d e +c^{2} d^{2}}{e^{3}}-\frac {d \left (b e -c d \right ) c}{e^{3}}-\frac {3 c b d}{5 e^{2}}-\frac {2 \left (\frac {c \left (2 b e -c d \right )}{e^{2}}-\frac {2 c \left (2 b e +2 c d \right )}{5 e^{2}}\right ) \left (b e +c d \right )}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \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 (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \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 )}\) | \(663\) |
| default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}\, \left (8 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c d \,e^{2}-24 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{2} d^{2} e +16 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{3} d^{3}+E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} e^{3}-17 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} c d \,e^{2}+32 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c^{2} d^{2} e -16 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{3} d^{3}-c^{4} e^{3} x^{4}-3 b \,c^{3} e^{3} x^{3}+2 c^{4} d \,e^{2} x^{3}-2 b^{2} c^{2} e^{3} x^{2}-5 b \,c^{3} d \,e^{2} x^{2}+8 c^{4} d^{2} e \,x^{2}-7 b^{2} c^{2} d \,e^{2} x +8 b \,c^{3} d^{2} e x \right )}{5 c^{2} x \left (c e \,x^{2}+b e x +c d x +b d \right ) e^{4}}\) | \(685\) |
1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(x*(e*x+d)*(c*x+b))^(1/2)/x/(c*x+b)*(2*( c*e*x^2+b*e*x)*d*(b*e-c*d)/e^4/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2/5*c/e^2*x *(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(c/e^2*(2*b*e-c*d)-2/5*c/e^2*(2 *b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(-d*(b^2*e^2-2*b* c*d*e+c^2*d^2)/e^4+d*(b*e-c*d)^2/e^4-b/e^3*d*(b*e-c*d)-1/3*(c/e^2*(2*b*e-c *d)-2/5*c/e^2*(2*b*e+2*c*d))/c/e*b*d)/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/( -1/c*b+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)*El lipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))+2*(1/e^3*(b^2*e ^2-2*b*c*d*e+c^2*d^2)-d*(b*e-c*d)/e^3*c-3/5*c/e^2*b*d-2/3*(c/e^2*(2*b*e-c* d)-2/5*c/e^2*(2*b*e+2*c*d))/c/e*(b*e+c*d))/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d /e)/(-1/c*b+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)*((-1/c*b+d/e)*EllipticE(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/ 2))-d/e*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.67 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left ({\left (16 \, c^{3} d^{4} - 24 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + b^{3} d e^{3} + {\left (16 \, c^{3} d^{3} e - 24 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} + b^{3} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (16 \, c^{3} d^{3} e - 16 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3} + {\left (16 \, c^{3} d^{2} e^{2} - 16 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (c^{3} e^{4} x^{2} - 8 \, c^{3} d^{2} e^{2} + 7 \, b c^{2} d e^{3} - 2 \, {\left (c^{3} d e^{3} - b c^{2} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{15 \, {\left (c^{2} e^{6} x + c^{2} d e^{5}\right )}} \]
-2/15*((16*c^3*d^4 - 24*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 + b^3*d*e^3 + (16*c^ 3*d^3*e - 24*b*c^2*d^2*e^2 + 6*b^2*c*d*e^3 + b^3*e^4)*x)*sqrt(c*e)*weierst rassPInverse(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*(16*c^3*d^3*e - 16*b*c^2*d^2*e^2 + b^2*c*d*e^3 + (16*c ^3*d^2*e^2 - 16*b*c^2*d*e^3 + b^2*c*e^4)*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), 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*(c^3*e ^4*x^2 - 8*c^3*d^2*e^2 + 7*b*c^2*d*e^3 - 2*(c^3*d*e^3 - b*c^2*e^4)*x)*sqrt (c*x^2 + b*x)*sqrt(e*x + d))/(c^2*e^6*x + c^2*d*e^5)
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]