Integrand size = 22, antiderivative size = 534 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\frac {2 \left (2 a \left (b^2-5 a c\right )+b \left (2 b^2-13 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{35 a^2 d^3 (d x)^{3/2}}-\frac {2 (5 a+3 b x) \left (a+b x+c x^2\right )^{3/2}}{35 a d (d x)^{7/2}}-\frac {\sqrt {2} b \left (b^2-8 a c\right ) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{35 a^2 \sqrt {c} d^{9/2} \sqrt {a+x (b+c x)}}+\frac {\sqrt {2} \sqrt {-b+\sqrt {b^2-4 a c}} \left (b^4-9 a b^2 c+20 a^2 c^2+b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{35 a^2 \sqrt {c} d^{9/2} \sqrt {a+x (b+c x)}} \] Output:
2/35*(2*a*(-5*a*c+b^2)+b*(-13*a*c+2*b^2)*x)*(c*x^2+b*x+a)^(1/2)/a^2/d^3/(d *x)^(3/2)-2/35*(3*b*x+5*a)*(c*x^2+b*x+a)^(3/2)/a/d/(d*x)^(7/2)-1/35*2^(1/2 )*b*(-8*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^(9/ 2)/(a+x*(c*x+b))^(1/2)+1/35*2^(1/2)*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b^4-9*a *b^2*c+20*a^2*c^2+b*(-8*a*c+b^2)*(-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)*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^(9/2)/(a+x*(c*x+b))^ (1/2)
Result contains complex when optimal does not.
Time = 12.33 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\frac {x \left (-4 b \left (b^2-8 a c\right ) x^3 (a+x (b+c x))-2 (a+x (b+c x)) \left (5 a^3-2 b^3 x^3+a^2 x (8 b+15 c x)+a b x^2 (b+16 c x)\right )+\frac {i b \left (b^2-8 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{9/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (b^4-9 a b^2 c+20 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{9/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{35 a^2 (d x)^{9/2} \sqrt {a+x (b+c x)}} \] Input:
Integrate[(a + b*x + c*x^2)^(3/2)/(d*x)^(9/2),x]
Output:
(x*(-4*b*(b^2 - 8*a*c)*x^3*(a + x*(b + c*x)) - 2*(a + x*(b + c*x))*(5*a^3 - 2*b^3*x^3 + a^2*x*(8*b + 15*c*x) + a*b*x^2*(b + 16*c*x)) + (I*b*(b^2 - 8 *a*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)] *x^(9/2)*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 - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] + (I*(b^4 - 9*a*b^2*c + 20*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a* b*c*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(9/2) *Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*Ellip ticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sq rt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] ))/(35*a^2*(d*x)^(9/2)*Sqrt[a + x*(b + c*x)])
Time = 1.18 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1161, 1229, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {3 \int \frac {(b+2 c x) \sqrt {c x^2+b x+a}}{(d x)^{7/2}}dx}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {3 \left (-\frac {2 \int \frac {d^2 \left (2 b \left (b^2-8 a c\right )+c \left (b^2-20 a c\right ) x\right )}{2 (d x)^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a d^4}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {2 b \left (b^2-8 a c\right )+c \left (b^2-20 a c\right ) x}{(d x)^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {3 \left (-\frac {-\frac {2 \int -\frac {c d \left (a \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right ) x\right )}{2 \sqrt {d x} \sqrt {c x^2+b x+a}}dx}{a d^2}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {c \int \frac {a \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right ) x}{\sqrt {d x} \sqrt {c x^2+b x+a}}dx}{a d}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 1241 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {c \sqrt {x} \int \frac {a \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{a d \sqrt {d x}}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {2 c \sqrt {x} \int \frac {a \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{a d \sqrt {d x}}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {2 c \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {2 \sqrt {a} b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {2 c \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {2 c \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {2 c \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {2 b \left (b^2-8 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {4 b \left (b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{15 a d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (x \left (10 a c+b^2\right )+3 a b\right )}{15 a d (d x)^{5/2}}\right )}{7 d}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 d (d x)^{7/2}}\) |
Input:
Int[(a + b*x + c*x^2)^(3/2)/(d*x)^(9/2),x]
Output:
(-2*(a + b*x + c*x^2)^(3/2))/(7*d*(d*x)^(7/2)) + (3*((-2*(3*a*b + (b^2 + 1 0*a*c)*x)*Sqrt[a + b*x + c*x^2])/(15*a*d*(d*x)^(5/2)) - ((-4*b*(b^2 - 8*a* c)*Sqrt[a + b*x + c*x^2])/(a*d*Sqrt[d*x]) + (2*c*Sqrt[x]*((-2*b*(b^2 - 8*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]*Ellip ticE[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]*(b^2 - 20*a*c) + (2*b*(b^2 - 8*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*Sq rt[d*x]))/(15*a*d^2)))/(7*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[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(899\) vs. \(2(444)=888\).
Time = 3.93 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.69
| method | result | size |
| elliptic | \(\frac {\sqrt {d x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 a \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{7 d^{5} x^{4}}-\frac {16 b \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{35 d^{5} x^{3}}-\frac {2 \left (15 a c +b^{2}\right ) \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{35 d^{5} a \,x^{2}}-\frac {4 \left (c d \,x^{2}+b d x +a d \right ) b \left (8 a c -b^{2}\right )}{35 d^{5} a^{2} \sqrt {x \left (c d \,x^{2}+b d x +a d \right )}}+\frac {\left (\frac {c^{2}}{d^{4}}-\frac {c \left (15 a c +b^{2}\right )}{35 a \,d^{4}}\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}}}}\, \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}}+\frac {2 b \left (8 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 )}{35 a^{2} d^{4} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right )}{\sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(900\) |
| risch | \(\text {Expression too large to display}\) | \(1076\) |
| default | \(\text {Expression too large to display}\) | \(1607\) |
Input:
int((c*x^2+b*x+a)^(3/2)/(d*x)^(9/2),x,method=_RETURNVERBOSE)
Output:
(d*x*(c*x^2+b*x+a))^(1/2)/(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/7/d^5*a*(c*d *x^3+b*d*x^2+a*d*x)^(1/2)/x^4-16/35/d^5*b*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)/x^ 3-2/35/d^5/a*(15*a*c+b^2)*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)/x^2-4/35*(c*d*x^2+ b*d*x+a*d)/d^5/a^2*b*(8*a*c-b^2)/(x*(c*d*x^2+b*d*x+a*d))^(1/2)+(c^2/d^4-1/ 35*c*(15*a*c+b^2)/a/d^4)*(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))+2/35*b*(8*a*c-b^2)/a^2/d^4*(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)*((-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...
Time = 0.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\frac {2 \, {\left ({\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} \sqrt {c d} x^{4} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 6 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} \sqrt {c d} x^{4} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - 3 \, {\left (8 \, a^{2} b c x + 5 \, a^{3} c - 2 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} x^{3} + {\left (a b^{2} c + 15 \, a^{2} c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{105 \, a^{2} c d^{5} x^{4}} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(d*x)^(9/2),x, algorithm="fricas")
Output:
2/105*((2*b^4 - 19*a*b^2*c + 60*a^2*c^2)*sqrt(c*d)*x^4*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^3*c - 8*a*b*c^2)*sqrt(c*d)*x^4*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*(8*a^2*b*c*x + 5*a^3*c - 2*(b^3*c - 8*a*b*c^2)*x^3 + (a*b^2*c + 15*a^2*c^2)*x^2)*sqrt(c*x^2 + b* x + a)*sqrt(d*x))/(a^2*c*d^5*x^4)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d x\right )^{\frac {9}{2}}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(3/2)/(d*x)**(9/2),x)
Output:
Integral((a + b*x + c*x**2)**(3/2)/(d*x)**(9/2), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{\left (d x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(d*x)^(9/2),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(3/2)/(d*x)^(9/2), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{\left (d x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(d*x)^(9/2),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^(3/2)/(d*x)^(9/2), x)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d\,x\right )}^{9/2}} \,d x \] Input:
int((a + b*x + c*x^2)^(3/2)/(d*x)^(9/2),x)
Output:
int((a + b*x + c*x^2)^(3/2)/(d*x)^(9/2), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d x)^{9/2}} \, dx=\frac {\sqrt {d}\, \left (-6 \sqrt {c \,x^{2}+b x +a}\, a -42 \sqrt {c \,x^{2}+b x +a}\, c \,x^{2}+24 \sqrt {x}\, \left (\int \frac {\sqrt {c \,x^{2}+b x +a}}{\sqrt {x}\, a \,x^{3}+\sqrt {x}\, b \,x^{4}+\sqrt {x}\, c \,x^{5}}d x \right ) a b \,x^{3}+20 \sqrt {x}\, \left (\int \frac {\sqrt {c \,x^{2}+b x +a}}{\sqrt {x}\, a \,x^{2}+\sqrt {x}\, b \,x^{3}+\sqrt {x}\, c \,x^{4}}d x \right ) a c \,x^{3}-56 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{5}+b \,x^{4}+a \,x^{3}}d x \right ) a c \,x^{3}+21 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{5}+b \,x^{4}+a \,x^{3}}d x \right ) b^{2} x^{3}\right )}{21 \sqrt {x}\, d^{5} x^{3}} \] Input:
int((c*x^2+b*x+a)^(3/2)/(d*x)^(9/2),x)
Output:
(sqrt(d)*( - 6*sqrt(a + b*x + c*x**2)*a - 42*sqrt(a + b*x + c*x**2)*c*x**2 + 24*sqrt(x)*int(sqrt(a + b*x + c*x**2)/(sqrt(x)*a*x**3 + sqrt(x)*b*x**4 + sqrt(x)*c*x**5),x)*a*b*x**3 + 20*sqrt(x)*int(sqrt(a + b*x + c*x**2)/(sqr t(x)*a*x**2 + sqrt(x)*b*x**3 + sqrt(x)*c*x**4),x)*a*c*x**3 - 56*sqrt(x)*in t((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x**3 + b*x**4 + c*x**5),x)*a*c*x**3 + 21*sqrt(x)*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x**3 + b*x**4 + c*x** 5),x)*b**2*x**3))/(21*sqrt(x)*d**5*x**3)