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

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

[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)

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Rubi [A]  time = 0.0500006, antiderivative size = 121, normalized size of antiderivative = 1., 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 = 0.0739362, 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))

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Maple [A]  time = 0.044, size = 174, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cx+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\,ab{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}cx+5\,{a}^{3}{c}^{3}+3\,{a}^{2}{b}^{2}{c}^{2}+6\,a{b}^{4}c-2\,{b}^{6} \right ) }{45\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{11}{2}}}} \end{align*}

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)

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Maxima [A]  time = 1.13138, size = 186, normalized size = 1.54 \begin{align*} \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} \end{align*}

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)

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Fricas [B]  time = 2.06714, size = 498, normalized size = 4.12 \begin{align*} \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 )}} \end{align*}

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)

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Sympy [A]  time = 24.1072, size = 1731, normalized size = 14.31 \begin{align*} \text{result too large to display} \end{align*}

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))

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Giac [A]  time = 1.21828, size = 238, normalized size = 1.97 \begin{align*} \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}} \end{align*}

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)