Integrand size = 22, antiderivative size = 504 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{15 a^2 d^3 \sqrt {d x}}-\frac {2 (3 a+b x) \sqrt {a+b x+c x^2}}{15 a d (d x)^{5/2}}-\frac {\sqrt {2} \left (b^2-3 a c\right ) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{15 a^2 \sqrt {c} d^{7/2} \sqrt {a+x (b+c x)}}+\frac {\sqrt {2} \sqrt {-b+\sqrt {b^2-4 a c}} \left (b^3-4 a b c+\sqrt {b^2-4 a c} \left (b^2-3 a c\right )\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{15 a^2 \sqrt {c} d^{7/2} \sqrt {a+x (b+c x)}} \] Output:
4/15*(-3*a*c+b^2)*(c*x^2+b*x+a)^(1/2)/a^2/d^3/(d*x)^(1/2)-2/15*(b*x+3*a)*( c*x^2+b*x+a)^(1/2)/a/d/(d*x)^(5/2)-1/15*2^(1/2)*(-3*a*c+b^2)*(-b+(-4*a*c+b ^2)^(1/2))^(1/2)*(b+(-4*a*c+b^2)^(1/2))*(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^( 1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(2^(1/2)*c^(1/2)*(d*x )^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2)/d^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+( -4*a*c+b^2)^(1/2)))^(1/2))/a^2/c^(1/2)/d^(7/2)/(a+x*(c*x+b))^(1/2)+1/15*2^ (1/2)*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b^3-4*a*b*c+(-4*a*c+b^2)^(1/2)*(-3*a* c+b^2))*(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1 /2)))^(1/2)*EllipticF(2^(1/2)*c^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^ (1/2)/d^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2))/a^2/c ^(1/2)/d^(7/2)/(a+x*(c*x+b))^(1/2)
Result contains complex when optimal does not.
Time = 23.81 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\frac {x \left (-4 \left (b^2-3 a c\right ) x^2 (a+x (b+c x))-2 (a+x (b+c x)) \left (3 a^2-2 b^2 x^2+a x (b+6 c x)\right )+\frac {i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{7/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (b^3-4 a b c-b^2 \sqrt {b^2-4 a c}+3 a c \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{7/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{15 a^2 (d x)^{7/2} \sqrt {a+x (b+c x)}} \] Input:
Integrate[Sqrt[a + b*x + c*x^2]/(d*x)^(7/2),x]
Output:
(x*(-4*(b^2 - 3*a*c)*x^2*(a + x*(b + c*x)) - 2*(a + x*(b + c*x))*(3*a^2 - 2*b^2*x^2 + a*x*(b + 6*c*x)) + (I*(b^2 - 3*a*c)*(-b + Sqrt[b^2 - 4*a*c])*S qrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(7/2)*Sqrt[(2*a + b*x - Sqrt[ b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]* Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sq rt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] + (I*(b^3 - 4*a*b*c - b ^2*Sqrt[b^2 - 4*a*c] + 3*a*c*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[ b^2 - 4*a*c])*x)]*x^(7/2)*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sq rt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4* a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a /(b + Sqrt[b^2 - 4*a*c])]))/(15*a^2*(d*x)^(7/2)*Sqrt[a + x*(b + c*x)])
Time = 1.09 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.86, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1161, 1237, 27, 1237, 27, 1241, 1240, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {\int \frac {b+2 c x}{(d x)^{5/2} \sqrt {c x^2+b x+a}}dx}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {2 \int \frac {d \left (2 \left (b^2-3 a c\right )+b c x\right )}{2 (d x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 a d^2}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {2 \left (b^2-3 a c\right )+b c x}{(d x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {-\frac {2 \int -\frac {c d \left (a b+2 \left (b^2-3 a c\right ) x\right )}{2 \sqrt {d x} \sqrt {c x^2+b x+a}}dx}{a d^2}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {c \int \frac {a b+2 \left (b^2-3 a c\right ) x}{\sqrt {d x} \sqrt {c x^2+b x+a}}dx}{a d}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 1241 |
| \(\displaystyle \frac {-\frac {\frac {c \sqrt {x} \int \frac {a b+2 \left (b^2-3 a c\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{a d \sqrt {d x}}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {-\frac {\frac {2 c \sqrt {x} \int \frac {a b+2 \left (b^2-3 a c\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{a d \sqrt {d x}}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {-\frac {\frac {2 c \sqrt {x} \left (\sqrt {a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {2 \sqrt {a} \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {2 c \sqrt {x} \left (\sqrt {a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {2 \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {-\frac {\frac {2 c \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {2 \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {-\frac {\frac {2 c \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {2 \left (b^2-3 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 \left (b^2-3 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 a d}-\frac {2 b \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}}{5 d}-\frac {2 \sqrt {a+b x+c x^2}}{5 d (d x)^{5/2}}\) |
Input:
Int[Sqrt[a + b*x + c*x^2]/(d*x)^(7/2),x]
Output:
(-2*Sqrt[a + b*x + c*x^2])/(5*d*(d*x)^(5/2)) + ((-2*b*Sqrt[a + b*x + c*x^2 ])/(3*a*d*(d*x)^(3/2)) - ((-4*(b^2 - 3*a*c)*Sqrt[a + b*x + c*x^2])/(a*d*Sq rt[d*x]) + (2*c*Sqrt[x]*((-2*(b^2 - 3*a*c)*(-((Sqrt[x]*Sqrt[a + b*x + c*x^ 2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^( 1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x + c*x^2])))/Sqr t[c] + (a^(1/4)*(Sqrt[a]*b + (2*(b^2 - 3*a*c))/Sqrt[c])*(Sqrt[a] + Sqrt[c] *x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^ (1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)*Sqrt[a + b*x + c*x^2])))/(a*d*Sqrt[d*x]))/(3*a*d))/(5*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)*((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[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_ )^2]), x_Symbol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(f + g*x)/(Sqrt[x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Time = 2.07 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.57
| method | result | size |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+b x +a}\, \left (6 a c \,x^{2}-2 b^{2} x^{2}+a b x +3 a^{2}\right )}{15 x^{2} a^{2} d^{3} \sqrt {d x}}-\frac {c \left (-\frac {\left (6 a c -2 b^{2}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {a b \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right ) \sqrt {d x \left (c \,x^{2}+b x +a \right )}}{15 a^{2} d^{3} \sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(790\) |
| elliptic | \(\frac {\sqrt {d x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{5 d^{4} x^{3}}-\frac {2 b \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{15 d^{4} a \,x^{2}}-\frac {4 \left (c d \,x^{2}+b d x +a d \right ) \left (3 a c -b^{2}\right )}{15 d^{4} a^{2} \sqrt {x \left (c d \,x^{2}+b d x +a d \right )}}-\frac {b \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{15 d^{3} a \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {2 \left (3 a c -b^{2}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{15 a^{2} d^{3} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right )}{\sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(842\) |
| default | \(\text {Expression too large to display}\) | \(1394\) |
Input:
int((c*x^2+b*x+a)^(1/2)/(d*x)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/15*(c*x^2+b*x+a)^(1/2)*(6*a*c*x^2-2*b^2*x^2+a*b*x+3*a^2)/x^2/a^2/d^3/(d *x)^(1/2)-1/15*c/a^2*(-(6*a*c-2*b^2)*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+ 1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(- b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^ 2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a *d*x)^(1/2)*((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))) *EllipticE(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2) )*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c -1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/2))*Elli pticF(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^ (1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/ c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))+a*b*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*( (x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c *(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c +b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^ 2+a*d*x)^(1/2)*EllipticF(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4* a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c +b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))/d^3*(d*x*(c*x^2+b*x +a))^(1/2)/(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\frac {2 \, {\left ({\left (2 \, b^{3} - 9 \, a b c\right )} \sqrt {c d} x^{3} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 6 \, {\left (b^{2} c - 3 \, a c^{2}\right )} \sqrt {c d} x^{3} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - 3 \, {\left (a b c x + 3 \, a^{2} c - 2 \, {\left (b^{2} c - 3 \, a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{45 \, a^{2} c d^{4} x^{3}} \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(d*x)^(7/2),x, algorithm="fricas")
Output:
2/45*((2*b^3 - 9*a*b*c)*sqrt(c*d)*x^3*weierstrassPInverse(4/3*(b^2 - 3*a*c )/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c) + 6*(b^2*c - 3*a*c^ 2)*sqrt(c*d)*x^3*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a *b*c)/c^3, weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b *c)/c^3, 1/3*(3*c*x + b)/c)) - 3*(a*b*c*x + 3*a^2*c - 2*(b^2*c - 3*a*c^2)* x^2)*sqrt(c*x^2 + b*x + a)*sqrt(d*x))/(a^2*c*d^4*x^3)
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d x\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(1/2)/(d*x)**(7/2),x)
Output:
Integral(sqrt(a + b*x + c*x**2)/(d*x)**(7/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{\left (d x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(d*x)^(7/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^2 + b*x + a)/(d*x)^(7/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{\left (d x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(d*x)^(7/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^2 + b*x + a)/(d*x)^(7/2), x)
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d\,x\right )}^{7/2}} \,d x \] Input:
int((a + b*x + c*x^2)^(1/2)/(d*x)^(7/2),x)
Output:
int((a + b*x + c*x^2)^(1/2)/(d*x)^(7/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{7/2}} \, dx=\frac {\sqrt {d}\, \left (-8 \sqrt {c \,x^{2}+b x +a}\, a -16 \sqrt {c \,x^{2}+b x +a}\, c \,x^{2}+4 \sqrt {x}\, \left (\int \frac {\sqrt {c \,x^{2}+b x +a}}{\sqrt {x}\, a \,x^{2}+\sqrt {x}\, b \,x^{3}+\sqrt {x}\, c \,x^{4}}d x \right ) a b \,x^{2}+3 \sqrt {x}\, \left (\int \frac {\sqrt {c \,x^{2}+b x +a}\, x}{\sqrt {x}\, a +\sqrt {x}\, b x +\sqrt {x}\, c \,x^{2}}d x \right ) c^{2} x^{2}+5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{2}+b x +a}d x \right ) c^{2} x^{2}\right )}{20 \sqrt {x}\, a \,d^{4} x^{2}} \] Input:
int((c*x^2+b*x+a)^(1/2)/(d*x)^(7/2),x)
Output:
(sqrt(d)*( - 8*sqrt(a + b*x + c*x**2)*a - 16*sqrt(a + b*x + c*x**2)*c*x**2 + 4*sqrt(x)*int(sqrt(a + b*x + c*x**2)/(sqrt(x)*a*x**2 + sqrt(x)*b*x**3 + sqrt(x)*c*x**4),x)*a*b*x**2 + 3*sqrt(x)*int((sqrt(a + b*x + c*x**2)*x)/(s qrt(x)*a + sqrt(x)*b*x + sqrt(x)*c*x**2),x)*c**2*x**2 + 5*sqrt(x)*int((sqr t(x)*sqrt(a + b*x + c*x**2))/(a + b*x + c*x**2),x)*c**2*x**2))/(20*sqrt(x) *a*d**4*x**2)