Integrand size = 27, antiderivative size = 140 \[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {697}{105} \sqrt {5-2 x} \sqrt {2+3 x+x^2}-\frac {1}{35} (5-2 x)^{3/2} (67+15 x) \sqrt {2+3 x+x^2}-\frac {5483 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{70 \sqrt {2+3 x+x^2}}+\frac {697 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{10 \sqrt {2+3 x+x^2}} \] Output:
697/105*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)-1/35*(5-2*x)^(3/2)*(67+15*x)*(x^2+ 3*x+2)^(1/2)-5483/70*(-x^2-3*x-2)^(1/2)*EllipticE((2+x)^(1/2),1/3*2^(1/2)) /(x^2+3*x+2)^(1/2)+697/10*(-x^2-3*x-2)^(1/2)*EllipticF((2+x)^(1/2),1/3*2^( 1/2))/(x^2+3*x+2)^(1/2)
Time = 29.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21 \[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {2 \sqrt {5-2 x} \left (-7886-14831 x-8638 x^2-1429 x^3+444 x^4+180 x^5\right )-16449 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} E\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right )|\frac {7}{9}\right )+1812 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right ),\frac {7}{9}\right )}{210 (-5+2 x) \sqrt {2+3 x+x^2}} \] Input:
Integrate[Sqrt[5 - 2*x]*(4 + 3*x)*Sqrt[2 + 3*x + x^2],x]
Output:
(2*Sqrt[5 - 2*x]*(-7886 - 14831*x - 8638*x^2 - 1429*x^3 + 444*x^4 + 180*x^ 5) - 16449*(5 - 2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*E llipticE[ArcSin[3/Sqrt[5 - 2*x]], 7/9] + 1812*(5 - 2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*EllipticF[ArcSin[3/Sqrt[5 - 2*x]], 7/9]) /(210*(-5 + 2*x)*Sqrt[2 + 3*x + x^2])
Time = 0.51 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1236, 27, 1231, 25, 1269, 1172, 27, 321, 327}
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.
\(\displaystyle \int \sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2} \, dx\) |
\(\Big \downarrow \) 1236 |
\(\displaystyle \frac {2}{7} \int \frac {(31 x+17) \sqrt {x^2+3 x+2}}{2 \sqrt {5-2 x}}dx+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \int \frac {(31 x+17) \sqrt {x^2+3 x+2}}{\sqrt {5-2 x}}dx+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle \frac {1}{7} \left (-\frac {1}{30} \int -\frac {5483 x+8248}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1}{15} \sqrt {5-2 x} \sqrt {x^2+3 x+2} (93 x+488)\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \int \frac {5483 x+8248}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {43911}{2} \int \frac {1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {5483}{2} \int \frac {\sqrt {5-2 x}}{\sqrt {x^2+3 x+2}}dx\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 1172 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {14637 \sqrt {-x^2-3 x-2} \int \frac {3}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {16449 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{3 \sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {43911 \sqrt {-x^2-3 x-2} \int \frac {1}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {5483 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {14637 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {5483 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {14637 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {16449 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\) |
Input:
Int[Sqrt[5 - 2*x]*(4 + 3*x)*Sqrt[2 + 3*x + x^2],x]
Output:
(6*Sqrt[5 - 2*x]*(2 + 3*x + x^2)^(3/2))/7 + (-1/15*(Sqrt[5 - 2*x]*(488 + 9 3*x)*Sqrt[2 + 3*x + x^2]) + ((-16449*Sqrt[-2 - 3*x - x^2]*EllipticE[ArcSin [Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2] + (14637*Sqrt[-2 - 3*x - x^2]*Ell ipticF[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2])/30)/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[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*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 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]
Time = 1.89 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}\, \left (720 x^{5}+790 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+5483 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+1776 x^{4}-5716 x^{3}-12620 x^{2}+6472 x +12320\right )}{840 x^{3}+420 x^{2}-4620 x -4200}\) | \(136\) |
elliptic | \(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (\frac {59 x \sqrt {-2 x^{3}-x^{2}+11 x +10}}{35}-\frac {44 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{15}-\frac {4124 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{735 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {5483 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{1470 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {6 x^{2} \sqrt {-2 x^{3}-x^{2}+11 x +10}}{7}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\) | \(228\) |
risch | \(-\frac {\left (90 x^{2}+177 x -308\right ) \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{105 \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}-\frac {\left (\frac {4124 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2205 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {5483 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{4410 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\) | \(235\) |
Input:
int((5-2*x)^(1/2)*(3*x+4)*(x^2+3*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/420*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)*(720*x^5+790*(5-2*x)^(1/2)*(14*x+14) ^(1/2)*(2*x+4)^(1/2)*EllipticF(1/3*(5-2*x)^(1/2),3/7*7^(1/2))+5483*(5-2*x) ^(1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)*EllipticE(1/3*(5-2*x)^(1/2),3/7*7^(1/ 2))+1776*x^4-5716*x^3-12620*x^2+6472*x+12320)/(2*x^3+x^2-11*x-10)
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.39 \[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {1}{105} \, {\left (90 \, x^{2} + 177 \, x - 308\right )} \sqrt {x^{2} + 3 \, x + 2} \sqrt {-2 \, x + 5} - \frac {8801}{252} \, \sqrt {-2} {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right ) + \frac {5483}{210} \, \sqrt {-2} {\rm weierstrassZeta}\left (\frac {67}{3}, \frac {440}{27}, {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right )\right ) \] Input:
integrate((5-2*x)^(1/2)*(4+3*x)*(x^2+3*x+2)^(1/2),x, algorithm="fricas")
Output:
1/105*(90*x^2 + 177*x - 308)*sqrt(x^2 + 3*x + 2)*sqrt(-2*x + 5) - 8801/252 *sqrt(-2)*weierstrassPInverse(67/3, 440/27, x + 1/6) + 5483/210*sqrt(-2)*w eierstrassZeta(67/3, 440/27, weierstrassPInverse(67/3, 440/27, x + 1/6))
\[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\int \sqrt {\left (x + 1\right ) \left (x + 2\right )} \sqrt {5 - 2 x} \left (3 x + 4\right )\, dx \] Input:
integrate((5-2*x)**(1/2)*(4+3*x)*(x**2+3*x+2)**(1/2),x)
Output:
Integral(sqrt((x + 1)*(x + 2))*sqrt(5 - 2*x)*(3*x + 4), x)
\[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\int { \sqrt {x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )} \sqrt {-2 \, x + 5} \,d x } \] Input:
integrate((5-2*x)^(1/2)*(4+3*x)*(x^2+3*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x^2 + 3*x + 2)*(3*x + 4)*sqrt(-2*x + 5), x)
\[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\int { \sqrt {x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )} \sqrt {-2 \, x + 5} \,d x } \] Input:
integrate((5-2*x)^(1/2)*(4+3*x)*(x^2+3*x+2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x^2 + 3*x + 2)*(3*x + 4)*sqrt(-2*x + 5), x)
Timed out. \[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\int \sqrt {5-2\,x}\,\left (3\,x+4\right )\,\sqrt {x^2+3\,x+2} \,d x \] Input:
int((5 - 2*x)^(1/2)*(3*x + 4)*(3*x + x^2 + 2)^(1/2),x)
Output:
int((5 - 2*x)^(1/2)*(3*x + 4)*(3*x + x^2 + 2)^(1/2), x)
\[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {6 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{7}+\frac {59 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{35}-\frac {2033 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{70}+\frac {5483 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{70}-\frac {25603 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{140} \] Input:
int((5-2*x)^(1/2)*(4+3*x)*(x^2+3*x+2)^(1/2),x)
Output:
(120*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**2 + 236*sqrt( - 2*x + 5)*sqr t(x**2 + 3*x + 2)*x - 4066*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2) + 10966*i nt((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**2)/(2*x**3 + x**2 - 11*x - 10 ),x) - 25603*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(2*x**3 + x**2 - 11*x - 10),x))/140