Integrand size = 23, antiderivative size = 428 \[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx=-\frac {2 (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x \sqrt {d+e x}}{105 c^2 e^2 \sqrt {b x+c x^2}}+\frac {2 \left (9 b d+\frac {3 c d^2}{e}-\frac {4 b^2 e}{c}\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c}+\frac {2 (3 c d+b e) x \sqrt {d+e x} \sqrt {b x+c x^2}}{35 c}+\frac {2}{7} x (d+e x)^{3/2} \sqrt {b x+c x^2}+\frac {2 \sqrt {b} (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 b^{3/2} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right ) \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{105 c^{5/2} e \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:
-2/105*(-b*e+2*c*d)*(8*b^2*e^2-3*b*c*d*e+3*c^2*d^2)*x*(e*x+d)^(1/2)/c^2/e^ 2/(c*x^2+b*x)^(1/2)+2/105*(9*b*d+3*c*d^2/e-4*b^2*e/c)*(e*x+d)^(1/2)*(c*x^2 +b*x)^(1/2)/c+2/35*(b*e+3*c*d)*x*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c+2/7*x*( e*x+d)^(3/2)*(c*x^2+b*x)^(1/2)+2/105*b^(1/2)*(-b*e+2*c*d)*(8*b^2*e^2-3*b*c *d*e+3*c^2*d^2)*x^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/2)*x^(1/2)/b^(1/2)/(1 +c*x/b)^(1/2),(1-b*e/c/d)^(1/2))/c^(5/2)/e^2/(b*(e*x+d)/d/(c*x+b))^(1/2)/( c*x^2+b*x)^(1/2)-2/105*b^(3/2)*(-4*b^2*e^2+9*b*c*d*e+3*c^2*d^2)*x^(1/2)*(e *x+d)^(1/2)*InverseJacobiAM(arctan(c^(1/2)*x^(1/2)/b^(1/2)),(1-b*e/c/d)^(1 /2))/c^(5/2)/e/(b*(e*x+d)/d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)
Result contains complex when optimal does not.
Time = 17.00 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx=\frac {2 \left (b e x (b+c x) (d+e x) \left (-4 b^2 e^2+3 b c e (3 d+e x)+3 c^2 \left (d^2+8 d e x+5 e^2 x^2\right )\right )-\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (6 c^3 d^3-9 b c^2 d^2 e+19 b^2 c d e^2-8 b^3 e^3\right ) (b+c x) (d+e x)+i b e \left (6 c^3 d^3-9 b c^2 d^2 e+19 b^2 c d e^2-8 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (3 c^3 d^3-18 b c^2 d^2 e+23 b^2 c d e^2-8 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{105 b c^2 e^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \] Input:
Integrate[(d + e*x)^(3/2)*Sqrt[b*x + c*x^2],x]
Output:
(2*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^2*e^2 + 3*b*c*e*(3*d + e*x) + 3*c^2*(d ^2 + 8*d*e*x + 5*e^2*x^2)) - Sqrt[b/c]*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2 *e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*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*b*e *(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*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)])))/(105*b*c^2*e^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])
Time = 1.00 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1166, 27, 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.
| \(\displaystyle \int \sqrt {b x+c x^2} (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {2 \int \frac {(d (7 c d-3 b e)+4 e (2 c d-b e) x) \sqrt {c x^2+b x}}{2 \sqrt {d+e x}}dx}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d (7 c d-3 b e)+4 e (2 c d-b e) x) \sqrt {c x^2+b x}}{\sqrt {d+e x}}dx}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {2 \int \frac {e \left (b d \left (3 c^2 d^2+9 b c e d-4 b^2 e^2\right )+(2 c d-b e) \left (3 c^2 d^2-3 b c e d+8 b^2 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {\int \frac {b d \left (3 c^2 d^2+9 b c e d-4 b^2 e^2\right )+(2 c d-b e) \left (3 c^2 d^2-3 b c e d+8 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {\frac {(2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {2 d (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 c e}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {\frac {\sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{15 c e}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e}}{7 c}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
Input:
Int[(d + e*x)^(3/2)*Sqrt[b*x + c*x^2],x]
Output:
(2*e*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(7*c) + ((2*Sqrt[d + e*x]*(3*c^2*d ^2 + 9*b*c*d*e - 4*b^2*e^2 + 12*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(1 5*c*e) - ((2*Sqrt[-b]*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sq rt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/S qrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (4*Sqrt[-b]*d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*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]))/(15*c*e))/( 7*c)
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[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[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]
Time = 0.93 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.55
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 e \,x^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7}+\frac {2 \left (b \,e^{2}+2 d e c -\frac {2 e \left (3 b e +3 c d \right )}{7}\right ) x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 c e}+\frac {2 \left (\frac {9 b d e}{7}+c \,d^{2}-\frac {2 \left (b \,e^{2}+2 d e c -\frac {2 e \left (3 b e +3 c d \right )}{7}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}-\frac {2 \left (\frac {9 b d e}{7}+c \,d^{2}-\frac {2 \left (b \,e^{2}+2 d e c -\frac {2 e \left (3 b e +3 c d \right )}{7}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) b \,d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{3 c \,e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (b \,d^{2}-\frac {3 \left (b \,e^{2}+2 d e c -\frac {2 e \left (3 b e +3 c d \right )}{7}\right ) b d}{5 c e}-\frac {2 \left (\frac {9 b d e}{7}+c \,d^{2}-\frac {2 \left (b \,e^{2}+2 d e c -\frac {2 e \left (3 b e +3 c d \right )}{7}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \left (b e +c d \right )}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \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 )}\) | \(664\) |
| default | \(-\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (-15 c^{4} e^{5} x^{5}+8 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{4} d \,e^{4}-23 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{3} c \,d^{2} e^{3}+18 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} c^{2} d^{3} e^{2}-3 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b \,c^{3} d^{4} e -8 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{4} d \,e^{4}+27 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{3} c \,d^{2} e^{3}-28 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} c^{2} d^{3} e^{2}+15 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b \,c^{3} d^{4} e -6 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{4} d^{5}-18 b \,c^{3} e^{5} x^{4}-39 c^{4} d \,e^{4} x^{4}+b^{2} c^{2} e^{5} x^{3}-51 b \,c^{3} d \,e^{4} x^{3}-27 c^{4} d^{2} e^{3} x^{3}+4 b^{3} c \,e^{5} x^{2}-8 b^{2} c^{2} d \,e^{4} x^{2}-36 b \,c^{3} d^{2} e^{3} x^{2}-3 c^{4} d^{3} e^{2} x^{2}+4 b^{3} c d \,e^{4} x -9 b^{2} c^{2} d^{2} e^{3} x -3 b \,c^{3} d^{3} e^{2} x \right )}{105 e^{3} x \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{3}}\) | \(920\) |
Input:
int((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((c*x+b)*x*(e*x+d))^(1/2)/x/(c*x+b)*(2/7 *e*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/5*(b*e^2+2*d*e*c-2/7*e*(3*b *e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(9/7*b*d*e+c*d^ 2-2/5*(b*e^2+2*d*e*c-2/7*e*(3*b*e+3*c*d))/c/e*(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)-2/3*(9/7*b*d*e+c*d^2-2/5*(b*e^2+2*d*e*c-2/7*e *(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e^2*b*d^2*((x+d/e)/d*e)^(1/2)*((b/c+x )/(-d/e+b/c))^(1/2)*(-e*x/d)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*E llipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))+2*(b*d^2-3/5*(b*e^2+ 2*d*e*c-2/7*e*(3*b*e+3*c*d))/c/e*b*d-2/3*(9/7*b*d*e+c*d^2-2/5*(b*e^2+2*d*e *c-2/7*e*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(b*e+c*d))*d/e*((x+d/e)/d*e )^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2 +b*d*x)^(1/2)*((-d/e+b/c)*EllipticE(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e+b/c))^ (1/2))-b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))))
Time = 0.21 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.09 \[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left ({\left (6 \, c^{4} d^{4} - 12 \, b c^{3} d^{3} e - 17 \, b^{2} c^{2} d^{2} e^{2} + 23 \, b^{3} c d e^{3} - 8 \, b^{4} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (6 \, c^{4} d^{3} e - 9 \, b c^{3} d^{2} e^{2} + 19 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (15 \, c^{4} e^{4} x^{2} + 3 \, c^{4} d^{2} e^{2} + 9 \, b c^{3} d e^{3} - 4 \, b^{2} c^{2} e^{4} + 3 \, {\left (8 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{315 \, c^{4} e^{3}} \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x, algorithm="fricas")
Output:
2/315*((6*c^4*d^4 - 12*b*c^3*d^3*e - 17*b^2*c^2*d^2*e^2 + 23*b^3*c*d*e^3 - 8*b^4*e^4)*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)) + 3*(6*c^4*d^3*e - 9*b*c^3*d^ 2*e^2 + 19*b^2*c^2*d*e^3 - 8*b^3*c*e^4)*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*(15*c^4*e^ 4*x^2 + 3*c^4*d^2*e^2 + 9*b*c^3*d*e^3 - 4*b^2*c^2*e^4 + 3*(8*c^4*d*e^3 + b *c^3*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^4*e^3)
\[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx=\int \sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {3}{2}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(c*x**2+b*x)**(1/2),x)
Output:
Integral(sqrt(x*(b + c*x))*(d + e*x)**(3/2), x)
\[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^2 + b*x)*(e*x + d)^(3/2), x)
\[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^2 + b*x)*(e*x + d)^(3/2), x)
Timed out. \[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx=\int \sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{3/2} \,d x \] Input:
int((b*x + c*x^2)^(1/2)*(d + e*x)^(3/2),x)
Output:
int((b*x + c*x^2)^(1/2)*(d + e*x)^(3/2), x)
\[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x)
Output:
( - 6*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*d*e + 4*sqrt(x)*sqrt(d + e* x)*sqrt(b + c*x)*b**2*e**2*x + 22*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c* d**2 + 36*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c*d*e*x + 20*sqrt(x)*sqrt( d + e*x)*sqrt(b + c*x)*b*c*e**2*x**2 + 32*sqrt(x)*sqrt(d + e*x)*sqrt(b + c *x)*c**2*d**2*x + 20*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**2*d*e*x**2 - 8 *int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c *d**2 + 2*b*c*d*e*x + b*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x**2),x)*b**4 *e**4 + 11*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b**2*d*e + b**2*e* *2*x + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x** 2),x)*b**3*c*d*e**3 + 10*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b**2 *d*e + b**2*e**2*x + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x**2),x)*b**2*c**2*d**2*e**2 - 3*int((sqrt(x)*sqrt(d + e*x)*sqr t(b + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2* x**2 + c**2*d**2*x + c**2*d*e*x**2),x)*b*c**3*d**3*e + 6*int((sqrt(x)*sqrt (d + e*x)*sqrt(b + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c*d*e* x + b*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x**2),x)*c**4*d**4 + 3*int((sqr t(x)*sqrt(d + e*x)*sqrt(b + c*x))/(b**2*d*e*x + b**2*e**2*x**2 + b*c*d**2* x + 2*b*c*d*e*x**2 + b*c*e**2*x**3 + c**2*d**2*x**2 + c**2*d*e*x**3),x)*b* *4*d**2*e**2 - 8*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x))/(b**2*d*e*x + b **2*e**2*x**2 + b*c*d**2*x + 2*b*c*d*e*x**2 + b*c*e**2*x**3 + c**2*d**2...