Integrand size = 23, antiderivative size = 392 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (128 c^2 d^2-112 b c d e+15 b^2 e^2+16 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+15 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
2/15*(6*c*e*x-5*b*e+16*c*d)*(c*x^2+b*x)^(3/2)/e^3/(e*x+d)^(3/2)-2/5*(c*x^2 +b*x)^(5/2)/e/(e*x+d)^(5/2)+4/15*(23*b^2*e^2-128*b*c*d*e+128*c^2*d^2)*Elli pticE(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)/e^6/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/1 5*(-b*e+2*c*d)*(15*b^2*e^2-128*b*c*d*e+128*c^2*d^2)*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^6/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/15*(128*c^2*d^2-112 *b*c*d*e+15*b^2*e^2+16*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^(1/2)/e^5/(e*x+d)^( 1/2)
Result contains complex when optimal does not.
Time = 23.68 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.02 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 (x (b+c x))^{5/2} \left (\frac {2 \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) (b+c x) (d+e x)}{\sqrt {x}}-\frac {e \sqrt {x} (b+c x) \left (b^2 e^2 \left (15 d^2+35 d e x+23 e^2 x^2\right )-b c e \left (112 d^3+256 d^2 e x+161 d e^2 x^2+11 e^3 x^3\right )+c^2 \left (128 d^4+288 d^3 e x+176 d^2 e^2 x^2+10 d e^3 x^3-3 e^4 x^4\right )\right )}{(d+e x)^2}+2 i \sqrt {\frac {b}{c}} c e \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i \sqrt {\frac {b}{c}} c e \left (128 c^2 d^2-144 b c d e+31 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{15 e^6 x^{5/2} (b+c x)^3 \sqrt {d+e x}} \]
(2*(x*(b + c*x))^(5/2)*((2*(128*c^2*d^2 - 128*b*c*d*e + 23*b^2*e^2)*(b + c *x)*(d + e*x))/Sqrt[x] - (e*Sqrt[x]*(b + c*x)*(b^2*e^2*(15*d^2 + 35*d*e*x + 23*e^2*x^2) - b*c*e*(112*d^3 + 256*d^2*e*x + 161*d*e^2*x^2 + 11*e^3*x^3) + c^2*(128*d^4 + 288*d^3*e*x + 176*d^2*e^2*x^2 + 10*d*e^3*x^3 - 3*e^4*x^4 )))/(d + e*x)^2 + (2*I)*Sqrt[b/c]*c*e*(128*c^2*d^2 - 128*b*c*d*e + 23*b^2* e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/S qrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(128*c^2*d^2 - 144*b*c*d*e + 31*b^ 2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c] /Sqrt[x]], (c*d)/(b*e)]))/(15*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])
Time = 0.62 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1161, 1230, 27, 1230, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {\int \frac {(b+2 c x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^{5/2}}dx}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int \frac {(b (16 c d-5 b e)+16 c (2 c d-b e) x) \sqrt {c x^2+b x}}{2 (d+e x)^{3/2}}dx}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\int \frac {(b (16 c d-5 b e)+16 c (2 c d-b e) x) \sqrt {c x^2+b x}}{(d+e x)^{3/2}}dx}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {2 \int \frac {b \left (128 c^2 d^2-112 b c e d+15 b^2 e^2\right )+2 c \left (128 c^2 d^2-128 b c e d+23 b^2 e^2\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {\int \frac {b \left (128 c^2 d^2-112 b c e d+15 b^2 e^2\right )+2 c \left (128 c^2 d^2-128 b c e d+23 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 c \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {(2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 c \sqrt {x} \sqrt {b+c x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 c \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
(-2*(b*x + c*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) + ((2*(16*c*d - 5*b*e + 6*c *e*x)*(b*x + c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) - ((2*(128*c^2*d^2 - 1 12*b*c*d*e + 15*b^2*e^2 + 16*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(3*e^ 2*Sqrt[d + e*x]) - ((4*Sqrt[-b]*Sqrt[c]*(128*c^2*d^2 - 128*b*c*d*e + 23*b^ 2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*S qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e + 15*b^2*e^2)*Sqrt[x ]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/S qrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]))/(3*e^ 2))/(5*e^2))/e
3.5.3.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (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]
Leaf count of result is larger than twice the leaf count of optimal. \(911\) vs. \(2(332)=664\).
Time = 2.03 (sec) , antiderivative size = 912, normalized size of antiderivative = 2.33
| 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 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{8} \left (x +\frac {d}{e}\right )^{3}}+\frac {22 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{15 e^{7} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+b e x \right ) \left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right )}{15 e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 c^{2} x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{4}}+\frac {2 \left (\frac {3 c^{2} \left (b e -c d \right )}{e^{4}}-\frac {2 c^{2} \left (2 b e +2 c d \right )}{5 e^{4}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (\frac {b^{3} e^{3}-9 b^{2} d \,e^{2} c +18 b \,c^{2} d^{2} e -10 c^{3} d^{3}}{e^{6}}+\frac {11 c d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right )}{15 e^{6}}-\frac {\left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \left (b e -c d \right )}{15 e^{6}}+\frac {b \left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right )}{15 e^{5}}-\frac {\left (\frac {3 c^{2} \left (b e -c d \right )}{e^{4}}-\frac {2 c^{2} \left (2 b e +2 c d \right )}{5 e^{4}}\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 {3 c \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right )}{e^{5}}+\frac {\left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) c}{15 e^{5}}-\frac {3 c^{2} b d}{5 e^{4}}-\frac {2 \left (\frac {3 c^{2} \left (b e -c d \right )}{e^{4}}-\frac {2 c^{2} \left (2 b e +2 c d \right )}{5 e^{4}}\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 )}\) | \(912\) |
| default | \(\text {Expression too large to display}\) | \(2170\) |
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/ 5*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/e^8*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2 )/(x+d/e)^3+22/15*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^7*(c*e*x^3+b*e*x^2+c*d *x^2+b*d*x)^(1/2)/(x+d/e)^2-2/15*(c*e*x^2+b*e*x)*(23*b^2*e^2-128*b*c*d*e+1 28*c^2*d^2)/e^6/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2/5*c^2/e^4*x*(c*e*x^3+b*e *x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(3*c^2/e^4*(b*e-c*d)-2/5*c^2/e^4*(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*((b^3*e^3-9*b^2*c*d*e^2+18 *b*c^2*d^2*e-10*c^3*d^3)/e^6+11/15*c*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^6-1 /15*(23*b^2*e^2-128*b*c*d*e+128*c^2*d^2)/e^6*(b*e-c*d)+1/15*b/e^5*(23*b^2* e^2-128*b*c*d*e+128*c^2*d^2)-1/3*(3*c^2/e^4*(b*e-c*d)-2/5*c^2/e^4*(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)*EllipticF(((1/c*b+x)*c/b )^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))+2*(3*c/e^5*(b^2*e^2-3*b*c*d*e+2*c^2*d ^2)+1/15*(23*b^2*e^2-128*b*c*d*e+128*c^2*d^2)/e^5*c-3/5*c^2/e^4*b*d-2/3*(3 *c^2/e^4*(b*e-c*d)-2/5*c^2/e^4*(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 = 0.36 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.07 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left ({\left (256 \, c^{3} d^{6} - 384 \, b c^{2} d^{5} e + 126 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + {\left (256 \, c^{3} d^{3} e^{3} - 384 \, b c^{2} d^{2} e^{4} + 126 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 3 \, {\left (256 \, c^{3} d^{4} e^{2} - 384 \, b c^{2} d^{3} e^{3} + 126 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (256 \, c^{3} d^{5} e - 384 \, b c^{2} d^{4} e^{2} + 126 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} 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 ) + 6 \, {\left (128 \, c^{3} d^{5} e - 128 \, b c^{2} d^{4} e^{2} + 23 \, b^{2} c d^{3} e^{3} + {\left (128 \, c^{3} d^{2} e^{4} - 128 \, b c^{2} d e^{5} + 23 \, b^{2} c e^{6}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{3} e^{3} - 128 \, b c^{2} d^{2} e^{4} + 23 \, b^{2} c d e^{5}\right )} x^{2} + 3 \, {\left (128 \, c^{3} d^{4} e^{2} - 128 \, b c^{2} d^{3} e^{3} + 23 \, b^{2} c d^{2} 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 (3 \, c^{3} e^{6} x^{4} - 128 \, c^{3} d^{4} e^{2} + 112 \, b c^{2} d^{3} e^{3} - 15 \, b^{2} c d^{2} e^{4} - {\left (10 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{3} - {\left (176 \, c^{3} d^{2} e^{4} - 161 \, b c^{2} d e^{5} + 23 \, b^{2} c e^{6}\right )} x^{2} - {\left (288 \, c^{3} d^{3} e^{3} - 256 \, b c^{2} d^{2} e^{4} + 35 \, b^{2} c d e^{5}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, {\left (c e^{10} x^{3} + 3 \, c d e^{9} x^{2} + 3 \, c d^{2} e^{8} x + c d^{3} e^{7}\right )}} \]
-2/45*((256*c^3*d^6 - 384*b*c^2*d^5*e + 126*b^2*c*d^4*e^2 + b^3*d^3*e^3 + (256*c^3*d^3*e^3 - 384*b*c^2*d^2*e^4 + 126*b^2*c*d*e^5 + b^3*e^6)*x^3 + 3* (256*c^3*d^4*e^2 - 384*b*c^2*d^3*e^3 + 126*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 3*(256*c^3*d^5*e - 384*b*c^2*d^4*e^2 + 126*b^2*c*d^3*e^3 + b^3*d^2*e^4)* x)*sqrt(c*e)*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)) + 6*(128*c^3*d^5*e - 128*b*c^2*d^4*e^2 + 23*b^2*c*d^3*e^3 + (128*c^3*d^2*e^4 - 128*b*c^2*d*e^5 + 23*b^2*c*e^6)*x ^3 + 3*(128*c^3*d^3*e^3 - 128*b*c^2*d^2*e^4 + 23*b^2*c*d*e^5)*x^2 + 3*(128 *c^3*d^4*e^2 - 128*b*c^2*d^3*e^3 + 23*b^2*c*d^2*e^4)*x)*sqrt(c*e)*weierstr assZeta(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*(3*c^3*e^6*x^4 - 128*c^3*d^4*e^2 + 112*b*c^2*d^3*e^3 - 15*b^2*c*d^2* e^4 - (10*c^3*d*e^5 - 11*b*c^2*e^6)*x^3 - (176*c^3*d^2*e^4 - 161*b*c^2*d*e ^5 + 23*b^2*c*e^6)*x^2 - (288*c^3*d^3*e^3 - 256*b*c^2*d^2*e^4 + 35*b^2*c*d *e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c*e^10*x^3 + 3*c*d*e^9*x^2 + 3* c*d^2*e^8*x + c*d^3*e^7)
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]