Integrand size = 26, antiderivative size = 84 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {32 c d^5 (b+2 c x)^2}{3 \sqrt {a+b x+c x^2}}+\frac {256}{3} c^2 d^5 \sqrt {a+b x+c x^2} \] Output:
-2/3*d^5*(2*c*x+b)^4/(c*x^2+b*x+a)^(3/2)-32/3*c*d^5*(2*c*x+b)^2/(c*x^2+b*x +a)^(1/2)+256/3*c^2*d^5*(c*x^2+b*x+a)^(1/2)
Time = 0.91 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {d^5 \left (-2 b^4-48 b^3 c x+192 b c^2 x \left (2 a+c x^2\right )+16 b^2 c \left (-2 a+3 c x^2\right )+32 c^2 \left (8 a^2+12 a c x^2+3 c^2 x^4\right )\right )}{3 (a+x (b+c x))^{3/2}} \] Input:
Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^(5/2),x]
Output:
(d^5*(-2*b^4 - 48*b^3*c*x + 192*b*c^2*x*(2*a + c*x^2) + 16*b^2*c*(-2*a + 3 *c*x^2) + 32*c^2*(8*a^2 + 12*a*c*x^2 + 3*c^2*x^4)))/(3*(a + x*(b + c*x))^( 3/2))
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1110, 27, 1110, 1104}
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.
\(\displaystyle \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1110 |
\(\displaystyle \frac {16}{3} c d^2 \int \frac {d^3 (b+2 c x)^3}{\left (c x^2+b x+a\right )^{3/2}}dx-\frac {2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {16}{3} c d^5 \int \frac {(b+2 c x)^3}{\left (c x^2+b x+a\right )^{3/2}}dx-\frac {2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1110 |
\(\displaystyle \frac {16}{3} c d^5 \left (8 c \int \frac {b+2 c x}{\sqrt {c x^2+b x+a}}dx-\frac {2 (b+2 c x)^2}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {16}{3} c d^5 \left (16 c \sqrt {a+b x+c x^2}-\frac {2 (b+2 c x)^2}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*d^5*(b + 2*c*x)^4)/(3*(a + b*x + c*x^2)^(3/2)) + (16*c*d^5*((-2*(b + 2 *c*x)^2)/Sqrt[a + b*x + c*x^2] + 16*c*Sqrt[a + b*x + c*x^2]))/3
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_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol ] :> Simp[d*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d*e*((m - 1)/(b*(p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^ 2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && N eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
Time = 1.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83
method | result | size |
risch | \(32 c^{2} d^{5} \sqrt {c \,x^{2}+b x +a}+\frac {2 \left (24 c^{2} x^{2}+24 c b x +20 a c +b^{2}\right ) \left (4 a c -b^{2}\right ) d^{5}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(70\) |
pseudoelliptic | \(\frac {256 d^{5} \left (\frac {3 c^{4} x^{4}}{8}+\frac {3 x^{2} \left (\frac {b x}{2}+a \right ) c^{3}}{2}+\left (\frac {3}{16} b^{2} x^{2}+\frac {3}{2} a b x +a^{2}\right ) c^{2}-\frac {\left (\frac {3 b x}{2}+a \right ) b^{2} c}{8}-\frac {b^{4}}{128}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(79\) |
gosper | \(\frac {2 d^{5} \left (48 c^{4} x^{4}+96 b \,c^{3} x^{3}+192 a \,c^{3} x^{2}+24 b^{2} c^{2} x^{2}+192 a b \,c^{2} x -24 b^{3} c x +128 a^{2} c^{2}-16 c a \,b^{2}-b^{4}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(91\) |
trager | \(\frac {2 d^{5} \left (48 c^{4} x^{4}+96 b \,c^{3} x^{3}+192 a \,c^{3} x^{2}+24 b^{2} c^{2} x^{2}+192 a b \,c^{2} x -24 b^{3} c x +128 a^{2} c^{2}-16 c a \,b^{2}-b^{4}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(91\) |
orering | \(\frac {2 \left (48 c^{4} x^{4}+96 b \,c^{3} x^{3}+192 a \,c^{3} x^{2}+24 b^{2} c^{2} x^{2}+192 a b \,c^{2} x -24 b^{3} c x +128 a^{2} c^{2}-16 c a \,b^{2}-b^{4}\right ) \left (2 c d x +b d \right )^{5}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (2 c x +b \right )^{5}}\) | \(107\) |
default | \(\text {Expression too large to display}\) | \(1936\) |
Input:
int((2*c*d*x+b*d)^5/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
32*c^2*d^5*(c*x^2+b*x+a)^(1/2)+2/3*(24*c^2*x^2+24*b*c*x+20*a*c+b^2)*(4*a*c -b^2)/(c*x^2+b*x+a)^(3/2)*d^5
Time = 0.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.67 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (48 \, c^{4} d^{5} x^{4} + 96 \, b c^{3} d^{5} x^{3} + 24 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{5} x^{2} - 24 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} d^{5} x - {\left (b^{4} + 16 \, a b^{2} c - 128 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \] Input:
integrate((2*c*d*x+b*d)^5/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/3*(48*c^4*d^5*x^4 + 96*b*c^3*d^5*x^3 + 24*(b^2*c^2 + 8*a*c^3)*d^5*x^2 - 24*(b^3*c - 8*a*b*c^2)*d^5*x - (b^4 + 16*a*b^2*c - 128*a^2*c^2)*d^5)*sqrt( c*x^2 + b*x + a)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)
Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (82) = 164\).
Time = 0.47 (sec) , antiderivative size = 702, normalized size of antiderivative = 8.36 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\begin {cases} \frac {256 a^{2} c^{2} d^{5}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {32 a b^{2} c d^{5}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {384 a b c^{2} d^{5} x}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {384 a c^{3} d^{5} x^{2}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {2 b^{4} d^{5}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {48 b^{3} c d^{5} x}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {48 b^{2} c^{2} d^{5} x^{2}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {192 b c^{3} d^{5} x^{3}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {96 c^{4} d^{5} x^{4}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} & \text {for}\: a \neq - x \left (b + c x\right ) \\\tilde {\infty } b^{5} d^{5} x + \tilde {\infty } b^{4} c d^{5} x^{2} + \tilde {\infty } b^{3} c^{2} d^{5} x^{3} + \tilde {\infty } b^{2} c^{3} d^{5} x^{4} + \tilde {\infty } b c^{4} d^{5} x^{5} + \tilde {\infty } c^{5} d^{5} x^{6} & \text {otherwise} \end {cases} \] Input:
integrate((2*c*d*x+b*d)**5/(c*x**2+b*x+a)**(5/2),x)
Output:
Piecewise((256*a**2*c**2*d**5/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) - 32*a*b**2*c*d**5/(3*a* sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) + 384*a*b*c**2*d**5*x/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*s qrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) + 384*a*c**3*d**5 *x**2/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x** 2*sqrt(a + b*x + c*x**2)) - 2*b**4*d**5/(3*a*sqrt(a + b*x + c*x**2) + 3*b* x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) - 48*b**3*c*d* *5*x/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2 *sqrt(a + b*x + c*x**2)) + 48*b**2*c**2*d**5*x**2/(3*a*sqrt(a + b*x + c*x* *2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) + 19 2*b*c**3*d**5*x**3/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x* *2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) + 96*c**4*d**5*x**4/(3*a*sqrt(a + b *x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x* *2)), Ne(a, -x*(b + c*x))), (zoo*b**5*d**5*x + zoo*b**4*c*d**5*x**2 + zoo* b**3*c**2*d**5*x**3 + zoo*b**2*c**3*d**5*x**4 + zoo*b*c**4*d**5*x**5 + zoo *c**5*d**5*x**6, True))
Exception generated. \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^5/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (72) = 144\).
Time = 0.37 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.60 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (24 \, {\left ({\left (2 \, {\left (\frac {{\left (b^{4} c^{6} d^{5} - 8 \, a b^{2} c^{7} d^{5} + 16 \, a^{2} c^{8} d^{5}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac {2 \, {\left (b^{5} c^{5} d^{5} - 8 \, a b^{3} c^{6} d^{5} + 16 \, a^{2} b c^{7} d^{5}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {b^{6} c^{4} d^{5} - 48 \, a^{2} b^{2} c^{6} d^{5} + 128 \, a^{3} c^{7} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac {b^{7} c^{3} d^{5} - 16 \, a b^{5} c^{4} d^{5} + 80 \, a^{2} b^{3} c^{5} d^{5} - 128 \, a^{3} b c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac {b^{8} c^{2} d^{5} + 8 \, a b^{6} c^{3} d^{5} - 240 \, a^{2} b^{4} c^{4} d^{5} + 1280 \, a^{3} b^{2} c^{5} d^{5} - 2048 \, a^{4} c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:
integrate((2*c*d*x+b*d)^5/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
2/3*(24*((2*((b^4*c^6*d^5 - 8*a*b^2*c^7*d^5 + 16*a^2*c^8*d^5)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 2*(b^5*c^5*d^5 - 8*a*b^3*c^6*d^5 + 16*a^2*b*c ^7*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (b^6*c^4*d^5 - 48*a^2*b^ 2*c^6*d^5 + 128*a^3*c^7*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x - (b^ 7*c^3*d^5 - 16*a*b^5*c^4*d^5 + 80*a^2*b^3*c^5*d^5 - 128*a^3*b*c^6*d^5)/(b^ 4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x - (b^8*c^2*d^5 + 8*a*b^6*c^3*d^5 - 24 0*a^2*b^4*c^4*d^5 + 1280*a^3*b^2*c^5*d^5 - 2048*a^4*c^6*d^5)/(b^4*c^2 - 8* a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)
Time = 6.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.40 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,b^4\,d^5+32\,a^2\,c^2\,d^5-96\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^2-16\,a\,b^2\,c\,d^5-192\,a\,c^2\,d^5\,\left (c\,x^2+b\,x+a\right )+48\,b^2\,c\,d^5\,\left (c\,x^2+b\,x+a\right )}{\sqrt {c\,x^2+b\,x+a}\,\left (3\,c\,x^2+3\,b\,x+3\,a\right )} \] Input:
int((b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^(5/2),x)
Output:
-(2*b^4*d^5 + 32*a^2*c^2*d^5 - 96*c^2*d^5*(a + b*x + c*x^2)^2 - 16*a*b^2*c *d^5 - 192*a*c^2*d^5*(a + b*x + c*x^2) + 48*b^2*c*d^5*(a + b*x + c*x^2))/( (a + b*x + c*x^2)^(1/2)*(3*a + 3*b*x + 3*c*x^2))
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.57 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c \,x^{2}+b x +a}\, d^{5} \left (48 c^{4} x^{4}+96 b \,c^{3} x^{3}+192 a \,c^{3} x^{2}+24 b^{2} c^{2} x^{2}+192 a b \,c^{2} x -24 b^{3} c x +128 a^{2} c^{2}-16 a \,b^{2} c -b^{4}\right )}{3 c^{2} x^{4}+6 b c \,x^{3}+6 a c \,x^{2}+3 b^{2} x^{2}+6 a b x +3 a^{2}} \] Input:
int((2*c*d*x+b*d)^5/(c*x^2+b*x+a)^(5/2),x)
Output:
(2*sqrt(a + b*x + c*x**2)*d**5*(128*a**2*c**2 - 16*a*b**2*c + 192*a*b*c**2 *x + 192*a*c**3*x**2 - b**4 - 24*b**3*c*x + 24*b**2*c**2*x**2 + 96*b*c**3* x**3 + 48*c**4*x**4))/(3*(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2*b*c* x**3 + c**2*x**4))