Integrand size = 14, antiderivative size = 197 \[ \int \left (c+b x+a x^2\right )^{5/2} \, dx=\frac {\sqrt {c+b x+a x^2} \left (15 b^5-160 a b^3 c+528 a^2 b c^2-10 a b^4 x+96 a^2 b^2 c x+1056 a^3 c^2 x+8 a^2 b^3 x^2+1248 a^3 b c x^2+432 a^3 b^2 x^3+832 a^4 c x^3+640 a^4 b x^4+256 a^5 x^5\right )}{1536 a^3}-\frac {5 \left (-b^6+12 a b^4 c-48 a^2 b^2 c^2+64 a^3 c^3\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{1024 a^{7/2}} \]
1/1536*(a*x^2+b*x+c)^(1/2)*(256*a^5*x^5+640*a^4*b*x^4+832*a^4*c*x^3+432*a^ 3*b^2*x^3+1248*a^3*b*c*x^2+8*a^2*b^3*x^2+1056*a^3*c^2*x+96*a^2*b^2*c*x-10* a*b^4*x+528*a^2*b*c^2-160*a*b^3*c+15*b^5)/a^3-5/1024*(64*a^3*c^3-48*a^2*b^ 2*c^2+12*a*b^4*c-b^6)*ln(b+2*a*x-2*a^(1/2)*(a*x^2+b*x+c)^(1/2))/a^(7/2)
Time = 1.37 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.76 \[ \int \left (c+b x+a x^2\right )^{5/2} \, dx=\frac {\sqrt {a} (b+2 a x) \sqrt {c+x (b+a x)} \left (15 b^4-40 a b^3 x+32 a^2 b x \left (13 c+8 a x^2\right )+8 a b^2 \left (-20 c+11 a x^2\right )+16 a^2 \left (33 c^2+26 a c x^2+8 a^2 x^4\right )\right )-15 \left (b^2-4 a c\right )^3 \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {c}+\sqrt {c+x (b+a x)}}\right )}{1536 a^{7/2}} \]
(Sqrt[a]*(b + 2*a*x)*Sqrt[c + x*(b + a*x)]*(15*b^4 - 40*a*b^3*x + 32*a^2*b *x*(13*c + 8*a*x^2) + 8*a*b^2*(-20*c + 11*a*x^2) + 16*a^2*(33*c^2 + 26*a*c *x^2 + 8*a^2*x^4)) - 15*(b^2 - 4*a*c)^3*ArcTanh[(Sqrt[a]*x)/(-Sqrt[c] + Sq rt[c + x*(b + a*x)])])/(1536*a^(7/2))
Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1087, 1087, 1087, 1092, 219}
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.
\(\displaystyle \int \left (a x^2+b x+c\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {(2 a x+b) \left (a x^2+b x+c\right )^{5/2}}{12 a}-\frac {5 \left (b^2-4 a c\right ) \int \left (a x^2+b x+c\right )^{3/2}dx}{24 a}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {(2 a x+b) \left (a x^2+b x+c\right )^{5/2}}{12 a}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(2 a x+b) \left (a x^2+b x+c\right )^{3/2}}{8 a}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {a x^2+b x+c}dx}{16 a}\right )}{24 a}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {(2 a x+b) \left (a x^2+b x+c\right )^{5/2}}{12 a}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(2 a x+b) \left (a x^2+b x+c\right )^{3/2}}{8 a}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(2 a x+b) \sqrt {a x^2+b x+c}}{4 a}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {a x^2+b x+c}}dx}{8 a}\right )}{16 a}\right )}{24 a}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {(2 a x+b) \left (a x^2+b x+c\right )^{5/2}}{12 a}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(2 a x+b) \left (a x^2+b x+c\right )^{3/2}}{8 a}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(2 a x+b) \sqrt {a x^2+b x+c}}{4 a}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(b+2 a x)^2}{a x^2+b x+c}}d\frac {b+2 a x}{\sqrt {a x^2+b x+c}}}{4 a}\right )}{16 a}\right )}{24 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(2 a x+b) \left (a x^2+b x+c\right )^{5/2}}{12 a}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(2 a x+b) \left (a x^2+b x+c\right )^{3/2}}{8 a}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(2 a x+b) \sqrt {a x^2+b x+c}}{4 a}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{8 a^{3/2}}\right )}{16 a}\right )}{24 a}\) |
((b + 2*a*x)*(c + b*x + a*x^2)^(5/2))/(12*a) - (5*(b^2 - 4*a*c)*(((b + 2*a *x)*(c + b*x + a*x^2)^(3/2))/(8*a) - (3*(b^2 - 4*a*c)*(((b + 2*a*x)*Sqrt[c + b*x + a*x^2])/(4*a) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqr t[c + b*x + a*x^2])])/(8*a^(3/2))))/(16*a)))/(24*a)
3.25.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Time = 0.33 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\left (2 a x +b \right ) \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}}}{12 a}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 a x +b \right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}}}{8 a}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 a x +b \right ) \sqrt {a \,x^{2}+b x +c}}{4 a}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{8 a^{\frac {3}{2}}}\right )}{16 a}\right )}{24 a}\) | \(143\) |
risch | \(\frac {\sqrt {a \,x^{2}+b x +c}\, \left (256 a^{5} x^{5}+640 a^{4} b \,x^{4}+832 a^{4} c \,x^{3}+432 a^{3} b^{2} x^{3}+1248 a^{3} b c \,x^{2}+8 a^{2} b^{3} x^{2}+1056 a^{3} c^{2} x +96 a^{2} b^{2} c x -10 a \,b^{4} x +528 a^{2} b \,c^{2}-160 a \,b^{3} c +15 b^{5}\right )}{1536 a^{3}}+\frac {5 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{1024 a^{\frac {7}{2}}}\) | \(187\) |
1/12*(2*a*x+b)/a*(a*x^2+b*x+c)^(5/2)+5/24*(4*a*c-b^2)/a*(1/8*(2*a*x+b)/a*( a*x^2+b*x+c)^(3/2)+3/16*(4*a*c-b^2)/a*(1/4*(2*a*x+b)*(a*x^2+b*x+c)^(1/2)/a +1/8*(4*a*c-b^2)/a^(3/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))))
Time = 0.28 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.16 \[ \int \left (c+b x+a x^2\right )^{5/2} \, dx=\left [-\frac {15 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {a} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - 4 \, \sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {a} - b^{2} - 4 \, a c\right ) - 4 \, {\left (256 \, a^{6} x^{5} + 640 \, a^{5} b x^{4} + 15 \, a b^{5} - 160 \, a^{2} b^{3} c + 528 \, a^{3} b c^{2} + 16 \, {\left (27 \, a^{4} b^{2} + 52 \, a^{5} c\right )} x^{3} + 8 \, {\left (a^{3} b^{3} + 156 \, a^{4} b c\right )} x^{2} - 2 \, {\left (5 \, a^{2} b^{4} - 48 \, a^{3} b^{2} c - 528 \, a^{4} c^{2}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{6144 \, a^{4}}, \frac {15 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {-a}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 2 \, {\left (256 \, a^{6} x^{5} + 640 \, a^{5} b x^{4} + 15 \, a b^{5} - 160 \, a^{2} b^{3} c + 528 \, a^{3} b c^{2} + 16 \, {\left (27 \, a^{4} b^{2} + 52 \, a^{5} c\right )} x^{3} + 8 \, {\left (a^{3} b^{3} + 156 \, a^{4} b c\right )} x^{2} - 2 \, {\left (5 \, a^{2} b^{4} - 48 \, a^{3} b^{2} c - 528 \, a^{4} c^{2}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{3072 \, a^{4}}\right ] \]
[-1/6144*(15*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(a)*log( -8*a^2*x^2 - 8*a*b*x - 4*sqrt(a*x^2 + b*x + c)*(2*a*x + b)*sqrt(a) - b^2 - 4*a*c) - 4*(256*a^6*x^5 + 640*a^5*b*x^4 + 15*a*b^5 - 160*a^2*b^3*c + 528* a^3*b*c^2 + 16*(27*a^4*b^2 + 52*a^5*c)*x^3 + 8*(a^3*b^3 + 156*a^4*b*c)*x^2 - 2*(5*a^2*b^4 - 48*a^3*b^2*c - 528*a^4*c^2)*x)*sqrt(a*x^2 + b*x + c))/a^ 4, 1/3072*(15*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-a)*ar ctan(1/2*sqrt(a*x^2 + b*x + c)*(2*a*x + b)*sqrt(-a)/(a^2*x^2 + a*b*x + a*c )) + 2*(256*a^6*x^5 + 640*a^5*b*x^4 + 15*a*b^5 - 160*a^2*b^3*c + 528*a^3*b *c^2 + 16*(27*a^4*b^2 + 52*a^5*c)*x^3 + 8*(a^3*b^3 + 156*a^4*b*c)*x^2 - 2* (5*a^2*b^4 - 48*a^3*b^2*c - 528*a^4*c^2)*x)*sqrt(a*x^2 + b*x + c))/a^4]
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (211) = 422\).
Time = 1.80 (sec) , antiderivative size = 1732, normalized size of antiderivative = 8.79 \[ \int \left (c+b x+a x^2\right )^{5/2} \, dx=\text {Too large to display} \]
a**2*Piecewise(((-b*(-3*b*(-5*b*(-b*c/(15*a) - 7*b*(c/6 - 3*b**2/(40*a))/( 8*a))/(6*a) - 3*c*(c/6 - 3*b**2/(40*a))/(4*a))/(4*a) - 2*c*(-b*c/(15*a) - 7*b*(c/6 - 3*b**2/(40*a))/(8*a))/(3*a))/(2*a) - c*(-5*b*(-b*c/(15*a) - 7*b *(c/6 - 3*b**2/(40*a))/(8*a))/(6*a) - 3*c*(c/6 - 3*b**2/(40*a))/(4*a))/(2* a))*Piecewise((log(2*sqrt(a)*sqrt(a*x**2 + b*x + c) + 2*a*x + b)/sqrt(a), Ne(c - b**2/(4*a), 0)), ((x + b/(2*a))*log(x + b/(2*a))/sqrt(a*(x + b/(2*a ))**2), True)) + sqrt(a*x**2 + b*x + c)*(x**5/6 + b*x**4/(60*a) + x**3*(c/ 6 - 3*b**2/(40*a))/(4*a) + x**2*(-b*c/(15*a) - 7*b*(c/6 - 3*b**2/(40*a))/( 8*a))/(3*a) + x*(-5*b*(-b*c/(15*a) - 7*b*(c/6 - 3*b**2/(40*a))/(8*a))/(6*a ) - 3*c*(c/6 - 3*b**2/(40*a))/(4*a))/(2*a) + (-3*b*(-5*b*(-b*c/(15*a) - 7* b*(c/6 - 3*b**2/(40*a))/(8*a))/(6*a) - 3*c*(c/6 - 3*b**2/(40*a))/(4*a))/(4 *a) - 2*c*(-b*c/(15*a) - 7*b*(c/6 - 3*b**2/(40*a))/(8*a))/(3*a))/a), Ne(a, 0)), (2*(c**4*(b*x + c)**(3/2)/3 - 4*c**3*(b*x + c)**(5/2)/5 + 6*c**2*(b* x + c)**(7/2)/7 - 4*c*(b*x + c)**(9/2)/9 + (b*x + c)**(11/2)/11)/b**5, Ne( b, 0)), (sqrt(c)*x**5/5, True)) + 2*a*b*Piecewise(((-b*(-3*b*(-3*b*c/(40*a ) - 5*b*(c/5 - 7*b**2/(80*a))/(6*a))/(4*a) - 2*c*(c/5 - 7*b**2/(80*a))/(3* a))/(2*a) - c*(-3*b*c/(40*a) - 5*b*(c/5 - 7*b**2/(80*a))/(6*a))/(2*a))*Pie cewise((log(2*sqrt(a)*sqrt(a*x**2 + b*x + c) + 2*a*x + b)/sqrt(a), Ne(c - b**2/(4*a), 0)), ((x + b/(2*a))*log(x + b/(2*a))/sqrt(a*(x + b/(2*a))**2), True)) + sqrt(a*x**2 + b*x + c)*(x**4/5 + b*x**3/(40*a) + x**2*(c/5 - ...
Exception generated. \[ \int \left (c+b x+a x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.30 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05 \[ \int \left (c+b x+a x^2\right )^{5/2} \, dx=\frac {1}{1536} \, \sqrt {a x^{2} + b x + c} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, a^{2} x + 5 \, a b\right )} x + \frac {27 \, a^{5} b^{2} + 52 \, a^{6} c}{a^{5}}\right )} x + \frac {a^{4} b^{3} + 156 \, a^{5} b c}{a^{5}}\right )} x - \frac {5 \, a^{3} b^{4} - 48 \, a^{4} b^{2} c - 528 \, a^{5} c^{2}}{a^{5}}\right )} x + \frac {15 \, a^{2} b^{5} - 160 \, a^{3} b^{3} c + 528 \, a^{4} b c^{2}}{a^{5}}\right )} + \frac {5 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x + c}\right )} \sqrt {a} + b \right |}\right )}{1024 \, a^{\frac {7}{2}}} \]
1/1536*sqrt(a*x^2 + b*x + c)*(2*(4*(2*(8*(2*a^2*x + 5*a*b)*x + (27*a^5*b^2 + 52*a^6*c)/a^5)*x + (a^4*b^3 + 156*a^5*b*c)/a^5)*x - (5*a^3*b^4 - 48*a^4 *b^2*c - 528*a^5*c^2)/a^5)*x + (15*a^2*b^5 - 160*a^3*b^3*c + 528*a^4*b*c^2 )/a^5) + 5/1024*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*log(abs(2 *(sqrt(a)*x - sqrt(a*x^2 + b*x + c))*sqrt(a) + b))/a^(7/2)
Time = 6.59 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.73 \[ \int \left (c+b x+a x^2\right )^{5/2} \, dx=\frac {\left (\frac {b}{2}+a\,x\right )\,{\left (a\,x^2+b\,x+c\right )}^{5/2}}{6\,a}+\frac {\left (5\,a\,c-\frac {5\,b^2}{4}\right )\,\left (\frac {\left (\left (\frac {x}{2}+\frac {b}{4\,a}\right )\,\sqrt {a\,x^2+b\,x+c}+\frac {\ln \left (\frac {\frac {b}{2}+a\,x}{\sqrt {a}}+\sqrt {a\,x^2+b\,x+c}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,a^{3/2}}\right )\,\left (3\,a\,c-\frac {3\,b^2}{4}\right )}{4\,a}+\frac {\left (\frac {b}{2}+a\,x\right )\,{\left (a\,x^2+b\,x+c\right )}^{3/2}}{4\,a}\right )}{6\,a} \]