### 3.1284 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^{13/2}} \, dx$$

Optimal. Leaf size=121 $-\frac{3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac{b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac{\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac{\sqrt{b d+2 c d x}}{64 c^4 d^7}$

[Out]

(b^2 - 4*a*c)^3/(704*c^4*d*(b*d + 2*c*d*x)^(11/2)) - (3*(b^2 - 4*a*c)^2)/(448*c^4*d^3*(b*d + 2*c*d*x)^(7/2)) +
(b^2 - 4*a*c)/(64*c^4*d^5*(b*d + 2*c*d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(64*c^4*d^7)

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Rubi [A]  time = 0.0533011, 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} $-\frac{3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac{b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac{\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac{\sqrt{b d+2 c d x}}{64 c^4 d^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(b^2 - 4*a*c)^3/(704*c^4*d*(b*d + 2*c*d*x)^(11/2)) - (3*(b^2 - 4*a*c)^2)/(448*c^4*d^3*(b*d + 2*c*d*x)^(7/2)) +
(b^2 - 4*a*c)/(64*c^4*d^5*(b*d + 2*c*d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(64*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)^{13/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{13/2}}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)^{9/2}}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^4 (b d+2 c d x)^{5/2}}+\frac{1}{64 c^3 d^6 \sqrt{b d+2 c d x}}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}-\frac{3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac{b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{64 c^4 d^7}\\ \end{align*}

Mathematica [A]  time = 0.0759858, size = 83, normalized size = 0.69 $\frac{77 \left (b^2-4 a c\right ) (b+2 c x)^4-33 \left (b^2-4 a c\right )^2 (b+2 c x)^2+7 \left (b^2-4 a c\right )^3+77 (b+2 c x)^6}{4928 c^4 d (d (b+2 c x))^{11/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(7*(b^2 - 4*a*c)^3 - 33*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 77*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 77*(b + 2*c*x)^6)/(49
28*c^4*d*(d*(b + 2*c*x))^(11/2))

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Maple [A]  time = 0.045, size = 174, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -77\,{c}^{6}{x}^{6}-231\,b{c}^{5}{x}^{5}+77\,a{c}^{5}{x}^{4}-308\,{b}^{2}{c}^{4}{x}^{4}+154\,ab{c}^{4}{x}^{3}-231\,{b}^{3}{c}^{3}{x}^{3}+33\,{a}^{2}{c}^{4}{x}^{2}+99\,a{b}^{2}{c}^{3}{x}^{2}-99\,{b}^{4}{c}^{2}{x}^{2}+33\,{a}^{2}b{c}^{3}x+22\,a{b}^{3}{c}^{2}x-22\,{b}^{5}cx+7\,{a}^{3}{c}^{3}+3\,{a}^{2}{b}^{2}{c}^{2}+2\,a{b}^{4}c-2\,{b}^{6} \right ) }{77\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{13}{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)^(13/2),x)

[Out]

-1/77*(2*c*x+b)*(-77*c^6*x^6-231*b*c^5*x^5+77*a*c^5*x^4-308*b^2*c^4*x^4+154*a*b*c^4*x^3-231*b^3*c^3*x^3+33*a^2
*c^4*x^2+99*a*b^2*c^3*x^2-99*b^4*c^2*x^2+33*a^2*b*c^3*x+22*a*b^3*c^2*x-22*b^5*c*x+7*a^3*c^3+3*a^2*b^2*c^2+2*a*
b^4*c-2*b^6)/c^4/(2*c*d*x+b*d)^(13/2)

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Maxima [A]  time = 1.25101, size = 186, normalized size = 1.54 \begin{align*} \frac{\frac{77 \, \sqrt{2 \, c d x + b d}}{c^{3} d^{6}} + \frac{77 \,{\left (2 \, c d x + b d\right )}^{4}{\left (b^{2} - 4 \, a c\right )} - 33 \,{\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} + 7 \,{\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{11}{2}} c^{3} d^{4}}}{4928 \, 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)^(13/2),x, algorithm="maxima")

[Out]

1/4928*(77*sqrt(2*c*d*x + b*d)/(c^3*d^6) + (77*(2*c*d*x + b*d)^4*(b^2 - 4*a*c) - 33*(b^4 - 8*a*b^2*c + 16*a^2*
c^2)*(2*c*d*x + b*d)^2*d^2 + 7*(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)^(11/2)*c
^3*d^4))/(c*d)

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Fricas [B]  time = 2.16869, size = 528, normalized size = 4.36 \begin{align*} \frac{{\left (77 \, c^{6} x^{6} + 231 \, b c^{5} x^{5} + 2 \, b^{6} - 2 \, a b^{4} c - 3 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 77 \,{\left (4 \, b^{2} c^{4} - a c^{5}\right )} x^{4} + 77 \,{\left (3 \, b^{3} c^{3} - 2 \, a b c^{4}\right )} x^{3} + 33 \,{\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - a^{2} c^{4}\right )} x^{2} + 11 \,{\left (2 \, b^{5} c - 2 \, a b^{3} c^{2} - 3 \, a^{2} b c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{77 \,{\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\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)^(13/2),x, algorithm="fricas")

[Out]

1/77*(77*c^6*x^6 + 231*b*c^5*x^5 + 2*b^6 - 2*a*b^4*c - 3*a^2*b^2*c^2 - 7*a^3*c^3 + 77*(4*b^2*c^4 - a*c^5)*x^4
+ 77*(3*b^3*c^3 - 2*a*b*c^4)*x^3 + 33*(3*b^4*c^2 - 3*a*b^2*c^3 - a^2*c^4)*x^2 + 11*(2*b^5*c - 2*a*b^3*c^2 - 3*
a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(64*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7
*x^3 + 60*b^4*c^6*d^7*x^2 + 12*b^5*c^5*d^7*x + b^6*c^4*d^7)

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Sympy [A]  time = 40.5587, size = 1975, normalized size = 16.32 \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)**(13/2),x)

[Out]

Piecewise((-7*a**3*c**3*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x*
*2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) -
3*a**2*b**2*c**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12
320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 33*a**2
*b*c**3*x*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**
3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 33*a**2*c**4*x
**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7
*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 2*a*b**4*c*sqrt(b*d
+ 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 +
18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 22*a*b**3*c**2*x*sqrt(b*d + 2*c*d
*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b
**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 99*a*b**2*c**3*x**2*sqrt(b*d + 2*c*d*x)/
(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*
c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 154*a*b*c**4*x**3*sqrt(b*d + 2*c*d*x)/(77*b*
*6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d
**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 77*a*c**5*x**4*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d
**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4
+ 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 2*b**6*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c*
*5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d*
*7*x**5 + 4928*c**10*d**7*x**6) + 22*b**5*c*x*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x +
4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 49
28*c**10*d**7*x**6) + 99*b**4*c**2*x**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b
**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**
10*d**7*x**6) + 231*b**3*c**3*x**3*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c
**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d*
*7*x**6) + 308*b**2*c**4*x**4*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d
**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x*
*6) + 231*b*c**5*x**5*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2
+ 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 77
*c**6*x**6*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b*
*3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6), 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)**(13/2), True))

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Giac [A]  time = 1.20978, size = 238, normalized size = 1.97 \begin{align*} \frac{\sqrt{2 \, c d x + b d}}{64 \, c^{4} d^{7}} + \frac{7 \, b^{6} d^{4} - 84 \, a b^{4} c d^{4} + 336 \, a^{2} b^{2} c^{2} d^{4} - 448 \, a^{3} c^{3} d^{4} - 33 \,{\left (2 \, c d x + b d\right )}^{2} b^{4} d^{2} + 264 \,{\left (2 \, c d x + b d\right )}^{2} a b^{2} c d^{2} - 528 \,{\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2} d^{2} + 77 \,{\left (2 \, c d x + b d\right )}^{4} b^{2} - 308 \,{\left (2 \, c d x + b d\right )}^{4} a c}{4928 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{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)^(13/2),x, algorithm="giac")

[Out]

1/64*sqrt(2*c*d*x + b*d)/(c^4*d^7) + 1/4928*(7*b^6*d^4 - 84*a*b^4*c*d^4 + 336*a^2*b^2*c^2*d^4 - 448*a^3*c^3*d^
4 - 33*(2*c*d*x + b*d)^2*b^4*d^2 + 264*(2*c*d*x + b*d)^2*a*b^2*c*d^2 - 528*(2*c*d*x + b*d)^2*a^2*c^2*d^2 + 77*
(2*c*d*x + b*d)^4*b^2 - 308*(2*c*d*x + b*d)^4*a*c)/((2*c*d*x + b*d)^(11/2)*c^4*d^5)