3.1283 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^{11/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac {3 \left (b^2-4 a c\right )}{64 c^4 d^5 \sqrt {b d+2 c d x}}-\frac {3 \left (b^2-4 a c\right )^2}{320 c^4 d^3 (b d+2 c d x)^{5/2}}+\frac {\left (b^2-4 a c\right )^3}{576 c^4 d (b d+2 c d x)^{9/2}}+\frac {(b d+2 c d x)^{3/2}}{192 c^4 d^7} \]

[Out]

1/576*(-4*a*c+b^2)^3/c^4/d/(2*c*d*x+b*d)^(9/2)-3/320*(-4*a*c+b^2)^2/c^4/d^3/(2*c*d*x+b*d)^(5/2)+1/192*(2*c*d*x
+b*d)^(3/2)/c^4/d^7+3/64*(-4*a*c+b^2)/c^4/d^5/(2*c*d*x+b*d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {683} \[ -\frac {3 \left (b^2-4 a c\right )^2}{320 c^4 d^3 (b d+2 c d x)^{5/2}}+\frac {3 \left (b^2-4 a c\right )}{64 c^4 d^5 \sqrt {b d+2 c d x}}+\frac {\left (b^2-4 a c\right )^3}{576 c^4 d (b d+2 c d x)^{9/2}}+\frac {(b d+2 c d x)^{3/2}}{192 c^4 d^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(11/2),x]

[Out]

(b^2 - 4*a*c)^3/(576*c^4*d*(b*d + 2*c*d*x)^(9/2)) - (3*(b^2 - 4*a*c)^2)/(320*c^4*d^3*(b*d + 2*c*d*x)^(5/2)) +
(3*(b^2 - 4*a*c))/(64*c^4*d^5*Sqrt[b*d + 2*c*d*x]) + (b*d + 2*c*d*x)^(3/2)/(192*c^4*d^7)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{11/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{11/2}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)^{7/2}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^4 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^3 d^6}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{576 c^4 d (b d+2 c d x)^{9/2}}-\frac {3 \left (b^2-4 a c\right )^2}{320 c^4 d^3 (b d+2 c d x)^{5/2}}+\frac {3 \left (b^2-4 a c\right )}{64 c^4 d^5 \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{192 c^4 d^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 83, normalized size = 0.69 \[ \frac {135 \left (b^2-4 a c\right ) (b+2 c x)^4-27 \left (b^2-4 a c\right )^2 (b+2 c x)^2+5 \left (b^2-4 a c\right )^3+15 (b+2 c x)^6}{2880 c^4 d (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(11/2),x]

[Out]

(5*(b^2 - 4*a*c)^3 - 27*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 135*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 15*(b + 2*c*x)^6)/(2
880*c^4*d*(d*(b + 2*c*x))^(9/2))

________________________________________________________________________________________

fricas [B]  time = 0.70, size = 238, normalized size = 1.97 \[ \frac {{\left (15 \, c^{6} x^{6} + 45 \, b c^{5} x^{5} + 2 \, b^{6} - 6 \, a b^{4} c - 3 \, a^{2} b^{2} c^{2} - 5 \, a^{3} c^{3} + 45 \, {\left (2 \, b^{2} c^{4} - 3 \, a c^{5}\right )} x^{4} + 15 \, {\left (7 \, b^{3} c^{3} - 18 \, a b c^{4}\right )} x^{3} + 9 \, {\left (7 \, b^{4} c^{2} - 21 \, a b^{2} c^{3} - 3 \, a^{2} c^{4}\right )} x^{2} + 9 \, {\left (2 \, b^{5} c - 6 \, a b^{3} c^{2} - 3 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{45 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(11/2),x, algorithm="fricas")

[Out]

1/45*(15*c^6*x^6 + 45*b*c^5*x^5 + 2*b^6 - 6*a*b^4*c - 3*a^2*b^2*c^2 - 5*a^3*c^3 + 45*(2*b^2*c^4 - 3*a*c^5)*x^4
+ 15*(7*b^3*c^3 - 18*a*b*c^4)*x^3 + 9*(7*b^4*c^2 - 21*a*b^2*c^3 - 3*a^2*c^4)*x^2 + 9*(2*b^5*c - 6*a*b^3*c^2 -
3*a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*
x^2 + 10*b^4*c^5*d^6*x + b^5*c^4*d^6)

________________________________________________________________________________________

giac [A]  time = 0.25, size = 176, normalized size = 1.45 \[ \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{192 \, c^{4} d^{7}} + \frac {5 \, b^{6} d^{4} - 60 \, a b^{4} c d^{4} + 240 \, a^{2} b^{2} c^{2} d^{4} - 320 \, a^{3} c^{3} d^{4} - 27 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} d^{2} + 216 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c d^{2} - 432 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2} d^{2} + 135 \, {\left (2 \, c d x + b d\right )}^{4} b^{2} - 540 \, {\left (2 \, c d x + b d\right )}^{4} a c}{2880 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c^{4} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(11/2),x, algorithm="giac")

[Out]

1/192*(2*c*d*x + b*d)^(3/2)/(c^4*d^7) + 1/2880*(5*b^6*d^4 - 60*a*b^4*c*d^4 + 240*a^2*b^2*c^2*d^4 - 320*a^3*c^3
*d^4 - 27*(2*c*d*x + b*d)^2*b^4*d^2 + 216*(2*c*d*x + b*d)^2*a*b^2*c*d^2 - 432*(2*c*d*x + b*d)^2*a^2*c^2*d^2 +
135*(2*c*d*x + b*d)^4*b^2 - 540*(2*c*d*x + b*d)^4*a*c)/((2*c*d*x + b*d)^(9/2)*c^4*d^5)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 174, normalized size = 1.44 \[ -\frac {\left (2 c x +b \right ) \left (-15 c^{6} x^{6}-45 b \,c^{5} x^{5}+135 a \,c^{5} x^{4}-90 b^{2} c^{4} x^{4}+270 a b \,c^{4} x^{3}-105 b^{3} c^{3} x^{3}+27 a^{2} c^{4} x^{2}+189 a \,b^{2} c^{3} x^{2}-63 b^{4} c^{2} x^{2}+27 a^{2} b \,c^{3} x +54 a \,b^{3} c^{2} x -18 b^{5} c x +5 a^{3} c^{3}+3 a^{2} b^{2} c^{2}+6 a \,b^{4} c -2 b^{6}\right )}{45 \left (2 c d x +b d \right )^{\frac {11}{2}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(11/2),x)

[Out]

-1/45*(2*c*x+b)*(-15*c^6*x^6-45*b*c^5*x^5+135*a*c^5*x^4-90*b^2*c^4*x^4+270*a*b*c^4*x^3-105*b^3*c^3*x^3+27*a^2*
c^4*x^2+189*a*b^2*c^3*x^2-63*b^4*c^2*x^2+27*a^2*b*c^3*x+54*a*b^3*c^2*x-18*b^5*c*x+5*a^3*c^3+3*a^2*b^2*c^2+6*a*
b^4*c-2*b^6)/c^4/(2*c*d*x+b*d)^(11/2)

________________________________________________________________________________________

maxima [A]  time = 1.40, size = 138, normalized size = 1.14 \[ \frac {\frac {15 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{c^{3} d^{6}} + \frac {135 \, {\left (2 \, c d x + b d\right )}^{4} {\left (b^{2} - 4 \, a c\right )} - 27 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{2} d^{2} + 5 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{4}}{{\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c^{3} d^{4}}}{2880 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(11/2),x, algorithm="maxima")

[Out]

1/2880*(15*(2*c*d*x + b*d)^(3/2)/(c^3*d^6) + (135*(2*c*d*x + b*d)^4*(b^2 - 4*a*c) - 27*(b^4 - 8*a*b^2*c + 16*a
^2*c^2)*(2*c*d*x + b*d)^2*d^2 + 5*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^4)/((2*c*d*x + b*d)^(9/2)
*c^3*d^4))/(c*d)

________________________________________________________________________________________

mupad [B]  time = 0.51, size = 147, normalized size = 1.21 \[ \frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}}{192\,c^4\,d^7}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (\frac {48\,a^2\,c^2\,d^2}{5}-\frac {24\,a\,b^2\,c\,d^2}{5}+\frac {3\,b^4\,d^2}{5}\right )+{\left (b\,d+2\,c\,d\,x\right )}^4\,\left (12\,a\,c-3\,b^2\right )-\frac {b^6\,d^4}{9}+\frac {64\,a^3\,c^3\,d^4}{9}-\frac {16\,a^2\,b^2\,c^2\,d^4}{3}+\frac {4\,a\,b^4\,c\,d^4}{3}}{64\,c^4\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(11/2),x)

[Out]

(b*d + 2*c*d*x)^(3/2)/(192*c^4*d^7) - ((b*d + 2*c*d*x)^2*((3*b^4*d^2)/5 + (48*a^2*c^2*d^2)/5 - (24*a*b^2*c*d^2
)/5) + (b*d + 2*c*d*x)^4*(12*a*c - 3*b^2) - (b^6*d^4)/9 + (64*a^3*c^3*d^4)/9 - (16*a^2*b^2*c^2*d^4)/3 + (4*a*b
^4*c*d^4)/3)/(64*c^4*d^5*(b*d + 2*c*d*x)^(9/2))

________________________________________________________________________________________

sympy [A]  time = 19.68, size = 1731, normalized size = 14.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(11/2),x)

[Out]

Piecewise((-5*a**3*c**3*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x*
*2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) - 3*a**2*b**2*c**2*sqrt(b*d + 2*c
*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b
*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) - 27*a**2*b*c**3*x*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c*
*5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5)
- 27*a**2*c**4*x**2*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2
+ 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) - 6*a*b**4*c*sqrt(b*d + 2*c*d*x)/(45
*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**
6*x**4 + 1440*c**9*d**6*x**5) - 54*a*b**3*c**2*x*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x
+ 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) - 189*a*
b**2*c**3*x**2*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600
*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) - 270*a*b*c**4*x**3*sqrt(b*d + 2*c*d*x)/(4
5*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d*
*6*x**4 + 1440*c**9*d**6*x**5) - 135*a*c**5*x**4*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x
+ 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) + 2*b**6
*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**
6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) + 18*b**5*c*x*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 4
50*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*
d**6*x**5) + 63*b**4*c**2*x**2*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*
d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) + 105*b**3*c**3*x**3*sqrt(
b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3
+ 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) + 90*b**2*c**4*x**4*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 4
50*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*
d**6*x**5) + 45*b*c**5*x**5*sqrt(b*d + 2*c*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**
6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b*c**8*d**6*x**4 + 1440*c**9*d**6*x**5) + 15*c**6*x**6*sqrt(b*d + 2*c
*d*x)/(45*b**5*c**4*d**6 + 450*b**4*c**5*d**6*x + 1800*b**3*c**6*d**6*x**2 + 3600*b**2*c**7*d**6*x**3 + 3600*b
*c**8*d**6*x**4 + 1440*c**9*d**6*x**5), Ne(c, 0)), ((a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4)/(b*
d)**(11/2), True))

________________________________________________________________________________________