Integrand size = 22, antiderivative size = 542 \[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \left (24 b^2-25 a c\right ) d^3 \sqrt {d x} \sqrt {a+b x+c x^2}}{105 c^3}-\frac {12 b d^2 (d x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {4 \sqrt {2} b \left (6 b^2-13 a c\right ) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) d^{7/2} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{105 c^{9/2} \sqrt {a+x (b+c x)}}+\frac {\sqrt {2} \sqrt {-b+\sqrt {b^2-4 a c}} \left (24 b^4-76 a b^2 c+25 a^2 c^2+4 b \left (6 b^2-13 a c\right ) \sqrt {b^2-4 a c}\right ) d^{7/2} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{105 c^{9/2} \sqrt {a+x (b+c x)}} \] Output:
2/105*(-25*a*c+24*b^2)*d^3*(d*x)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3-12/35*b*d^2 *(d*x)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2+2/7*d*(d*x)^(5/2)*(c*x^2+b*x+a)^(1/2) /c-4/105*2^(1/2)*b*(-13*a*c+6*b^2)*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b+(-4*a* c+b^2)^(1/2))*d^(7/2)*(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))/c^(9/2)/(a+x*(c*x+b))^(1/2)+1/105*2^(1/2)*(-b+(-4*a*c+b^2)^(1/2)) ^(1/2)*(24*b^4-76*a*b^2*c+25*a^2*c^2+4*b*(-13*a*c+6*b^2)*(-4*a*c+b^2)^(1/2 ))*d^(7/2)*(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))/c^ (9/2)/(a+x*(c*x+b))^(1/2)
Result contains complex when optimal does not.
Time = 24.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.99 \[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {(d x)^{7/2} \left (\frac {2 \sqrt {x} (a+x (b+c x)) \left (24 b^2-18 b c x+5 c \left (-5 a+3 c x^2\right )\right )}{c^3}+\frac {x \left (-\frac {16 b \left (6 b^2-13 a c\right ) (a+x (b+c x))}{x^{3/2}}+\frac {4 i b \left (6 b^2-13 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (24 b^4-76 a b^2 c+25 a^2 c^2-24 b^3 \sqrt {b^2-4 a c}+52 a b c \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{c^4}\right )}{105 x^{7/2} \sqrt {a+x (b+c x)}} \] Input:
Integrate[(d*x)^(7/2)/Sqrt[a + b*x + c*x^2],x]
Output:
((d*x)^(7/2)*((2*Sqrt[x]*(a + x*(b + c*x))*(24*b^2 - 18*b*c*x + 5*c*(-5*a + 3*c*x^2)))/c^3 + (x*((-16*b*(6*b^2 - 13*a*c)*(a + x*(b + c*x)))/x^(3/2) + ((4*I)*b*(6*b^2 - 13*a*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*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 - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] + (I*(24*b^4 - 76*a*b^2*c + 25*a^2*c^2 - 24*b^3*Sqr t[b^2 - 4*a*c] + 52*a*b*c*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4* a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqr t[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[ b^2 - 4*a*c])]))/c^4))/(105*x^(7/2)*Sqrt[a + x*(b + c*x)])
Time = 1.19 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.82, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1166, 27, 1236, 27, 1236, 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.
| \(\displaystyle \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {2 \int -\frac {d^2 (d x)^{3/2} (5 a+6 b x)}{2 \sqrt {c x^2+b x+a}}dx}{7 c}+\frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \int \frac {(d x)^{3/2} (5 a+6 b x)}{\sqrt {c x^2+b x+a}}dx}{7 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {2 \int -\frac {d \sqrt {d x} \left (18 a b+\left (24 b^2-25 a c\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{5 c}+\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \int \frac {\sqrt {d x} \left (18 a b+\left (24 b^2-25 a c\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \int -\frac {d \left (a \left (24 b^2-25 a c\right )+8 b \left (6 b^2-13 a c\right ) x\right )}{2 \sqrt {d x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}\right )}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}-\frac {d \int \frac {a \left (24 b^2-25 a c\right )+8 b \left (6 b^2-13 a c\right ) x}{\sqrt {d x} \sqrt {c x^2+b x+a}}dx}{3 c}\right )}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 1241 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}-\frac {d \sqrt {x} \int \frac {a \left (24 b^2-25 a c\right )+8 b \left (6 b^2-13 a c\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{3 c \sqrt {d x}}\right )}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}-\frac {2 d \sqrt {x} \int \frac {a \left (24 b^2-25 a c\right )+8 b \left (6 b^2-13 a c\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{3 c \sqrt {d x}}\right )}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}-\frac {2 d \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} \left (24 b^2-25 a c\right )+\frac {8 b \left (6 b^2-13 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {8 \sqrt {a} b \left (6 b^2-13 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 c \sqrt {d x}}\right )}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}-\frac {2 d \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} \left (24 b^2-25 a c\right )+\frac {8 b \left (6 b^2-13 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {8 b \left (6 b^2-13 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 c \sqrt {d x}}\right )}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}-\frac {2 d \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (24 b^2-25 a c\right )+\frac {8 b \left (6 b^2-13 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {8 b \left (6 b^2-13 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 c \sqrt {d x}}\right )}{5 c}\right )}{7 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 d (d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}-\frac {d^2 \left (\frac {12 b (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {d \left (\frac {2 \sqrt {d x} \left (24 b^2-25 a c\right ) \sqrt {a+b x+c x^2}}{3 c}-\frac {2 d \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (24 b^2-25 a c\right )+\frac {8 b \left (6 b^2-13 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {8 b \left (6 b^2-13 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{3 c \sqrt {d x}}\right )}{5 c}\right )}{7 c}\) |
Input:
Int[(d*x)^(7/2)/Sqrt[a + b*x + c*x^2],x]
Output:
(2*d*(d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c) - (d^2*((12*b*(d*x)^(3/2)*Sq rt[a + b*x + c*x^2])/(5*c) - (d*((2*(24*b^2 - 25*a*c)*Sqrt[d*x]*Sqrt[a + b *x + c*x^2])/(3*c) - (2*d*Sqrt[x]*((-8*b*(6*b^2 - 13*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])))/Sqrt[c] + (a^(1/4)*(Sqrt[a]*(24*b^2 - 25*a*c) + (8*b*(6*b^2 - 13*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])))/(3*c*Sqrt[d*x])))/(5* c)))/(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[((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_)*((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[((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 = 2.79 (sec) , antiderivative size = 889, normalized size of antiderivative = 1.64
| method | result | size |
| elliptic | \(\frac {\sqrt {d x}\, \sqrt {d x \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 d^{3} x^{2} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{7 c}-\frac {12 d^{3} b x \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{35 c^{2}}+\frac {2 \left (-\frac {5 d^{4} a}{7 c}+\frac {24 d^{4} b^{2}}{35 c^{2}}\right ) \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{3 c d}-\frac {\left (-\frac {5 d^{4} a}{7 c}+\frac {24 d^{4} b^{2}}{35 c^{2}}\right ) a \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 )}{3 c^{2} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {\left (\frac {18 d^{4} b a}{35 c^{2}}-\frac {2 \left (-\frac {5 d^{4} a}{7 c}+\frac {24 d^{4} b^{2}}{35 c^{2}}\right ) b}{3 c}\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}}\right )}{d x \sqrt {c \,x^{2}+b x +a}}\) | \(889\) |
| risch | \(\text {Expression too large to display}\) | \(1050\) |
| default | \(\text {Expression too large to display}\) | \(1543\) |
Input:
int((d*x)^(7/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/d/x*(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)*(d*x*(c*x^2+b*x+a))^(1/2)*(2/7*d^3/c *x^2*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)-12/35*d^3/c^2*b*x*(c*d*x^3+b*d*x^2+a*d* x)^(1/2)+2/3*(-5/7*d^4/c*a+24/35*d^4/c^2*b^2)/c/d*(c*d*x^3+b*d*x^2+a*d*x)^ (1/2)-1/3*(-5/7*d^4/c*a+24/35*d^4/c^2*b^2)/c^2*a*(b+(-4*a*c+b^2)^(1/2))*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))+(18/35*d^4/c^ 2*b*a-2/3*(-5/7*d^4/c*a+24/35*d^4/c^2*b^2)/c*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)*((-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))*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)...
Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.43 \[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (48 \, b^{4} - 176 \, a b^{2} c + 75 \, a^{2} c^{2}\right )} \sqrt {c d} d^{3} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 24 \, {\left (6 \, b^{3} c - 13 \, a b c^{2}\right )} \sqrt {c d} d^{3} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (15 \, c^{4} d^{3} x^{2} - 18 \, b c^{3} d^{3} x + {\left (24 \, b^{2} c^{2} - 25 \, a c^{3}\right )} d^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{315 \, c^{5}} \] Input:
integrate((d*x)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/315*((48*b^4 - 176*a*b^2*c + 75*a^2*c^2)*sqrt(c*d)*d^3*weierstrassPInver se(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) + 24*(6*b^3*c - 13*a*b*c^2)*sqrt(c*d)*d^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*(15*c^4*d^3*x^2 - 18*b*c^3*d^3*x + (24*b^2*c^2 - 25*a*c^3)*d^3)*sqrt(c*x^2 + b*x + a)*sqrt (d*x))/c^5
\[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d x\right )^{\frac {7}{2}}}{\sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((d*x)**(7/2)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((d*x)**(7/2)/sqrt(a + b*x + c*x**2), x)
\[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\left (d x\right )^{\frac {7}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:
integrate((d*x)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x)^(7/2)/sqrt(c*x^2 + b*x + a), x)
\[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\left (d x\right )^{\frac {7}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:
integrate((d*x)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x)^(7/2)/sqrt(c*x^2 + b*x + a), x)
Timed out. \[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (d\,x\right )}^{7/2}}{\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((d*x)^(7/2)/(a + b*x + c*x^2)^(1/2),x)
Output:
int((d*x)^(7/2)/(a + b*x + c*x^2)^(1/2), x)
\[ \int \frac {(d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {d}\, d^{3} \left (18 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a -12 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b x +10 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, c \,x^{2}-52 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a c +24 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) b^{2}-9 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{2}\right )}{35 c^{2}} \] Input:
int((d*x)^(7/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
(sqrt(d)*d**3*(18*sqrt(x)*sqrt(a + b*x + c*x**2)*a - 12*sqrt(x)*sqrt(a + b *x + c*x**2)*b*x + 10*sqrt(x)*sqrt(a + b*x + c*x**2)*c*x**2 - 52*int((sqrt (x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x)*a*c + 24*int((sqrt(x)* sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x)*b**2 - 9*int((sqrt(x)*sqrt (a + b*x + c*x**2))/(a*x + b*x**2 + c*x**3),x)*a**2))/(35*c**2)