### 3.1279 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^{3/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.049329, 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} $-\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{448 c^4 d^5}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{64 c^4 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{11/2}}{704 c^4 d^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0681235, size = 83, normalized size = 0.69 $\frac{-33 \left (b^2-4 a c\right ) (b+2 c x)^4+77 \left (b^2-4 a c\right )^2 (b+2 c x)^2+77 \left (b^2-4 a c\right )^3+7 (b+2 c x)^6}{4928 c^4 d \sqrt{d (b+2 c x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 173, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -7\,{c}^{6}{x}^{6}-21\,b{c}^{5}{x}^{5}-33\,a{c}^{5}{x}^{4}-18\,{b}^{2}{c}^{4}{x}^{4}-66\,ab{c}^{4}{x}^{3}-{b}^{3}{c}^{3}{x}^{3}-77\,{a}^{2}{c}^{4}{x}^{2}-11\,a{b}^{2}{c}^{3}{x}^{2}+{b}^{4}{c}^{2}{x}^{2}-77\,{a}^{2}b{c}^{3}x+22\,a{b}^{3}{c}^{2}x-2\,{b}^{5}cx+77\,{a}^{3}{c}^{3}-77\,{a}^{2}{b}^{2}{c}^{2}+22\,a{b}^{4}c-2\,{b}^{6} \right ) }{77\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

-1/77*(2*c*x+b)*(-7*c^6*x^6-21*b*c^5*x^5-33*a*c^5*x^4-18*b^2*c^4*x^4-66*a*b*c^4*x^3-b^3*c^3*x^3-77*a^2*c^4*x^2
-11*a*b^2*c^3*x^2+b^4*c^2*x^2-77*a^2*b*c^3*x+22*a*b^3*c^2*x-2*b^5*c*x+77*a^3*c^3-77*a^2*b^2*c^2+22*a*b^4*c-2*b
^6)/c^4/(2*c*d*x+b*d)^(3/2)

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Maxima [A]  time = 1.07999, size = 184, normalized size = 1.52 \begin{align*} \frac{\frac{77 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}}{\sqrt{2 \, c d x + b d} c^{3}} - \frac{33 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 77 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} d^{4} - 7 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{c^{3} d^{6}}}{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)^(3/2),x, algorithm="maxima")

[Out]

1/4928*(77*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/(sqrt(2*c*d*x + b*d)*c^3) - (33*(2*c*d*x + b*d)^(7
/2)*(b^2 - 4*a*c)*d^2 - 77*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^(3/2)*d^4 - 7*(2*c*d*x + b*d)^(11/2)
)/(c^3*d^6))/(c*d)

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Fricas [A]  time = 2.06958, size = 375, normalized size = 3.1 \begin{align*} \frac{{\left (7 \, c^{6} x^{6} + 21 \, b c^{5} x^{5} + 2 \, b^{6} - 22 \, a b^{4} c + 77 \, a^{2} b^{2} c^{2} - 77 \, a^{3} c^{3} + 3 \,{\left (6 \, b^{2} c^{4} + 11 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} + 66 \, a b c^{4}\right )} x^{3} -{\left (b^{4} c^{2} - 11 \, a b^{2} c^{3} - 77 \, a^{2} c^{4}\right )} x^{2} +{\left (2 \, b^{5} c - 22 \, a b^{3} c^{2} + 77 \, a^{2} b c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{77 \,{\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\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)^(3/2),x, algorithm="fricas")

[Out]

1/77*(7*c^6*x^6 + 21*b*c^5*x^5 + 2*b^6 - 22*a*b^4*c + 77*a^2*b^2*c^2 - 77*a^3*c^3 + 3*(6*b^2*c^4 + 11*a*c^5)*x
^4 + (b^3*c^3 + 66*a*b*c^4)*x^3 - (b^4*c^2 - 11*a*b^2*c^3 - 77*a^2*c^4)*x^2 + (2*b^5*c - 22*a*b^3*c^2 + 77*a^2
*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(2*c^5*d^2*x + b*c^4*d^2)

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Sympy [A]  time = 68.8422, size = 128, normalized size = 1.06 \begin{align*} - \frac{\left (4 a c - b^{2}\right )^{3}}{64 c^{4} d \sqrt{b d + 2 c d x}} + \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{192 c^{4} d^{3}} + \frac{\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}}{448 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{11}{2}}}{704 c^{4} d^{7}} \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)**(3/2),x)

[Out]

-(4*a*c - b**2)**3/(64*c**4*d*sqrt(b*d + 2*c*d*x)) + (b*d + 2*c*d*x)**(3/2)*(48*a**2*c**2 - 24*a*b**2*c + 3*b*
*4)/(192*c**4*d**3) + (12*a*c - 3*b**2)*(b*d + 2*c*d*x)**(7/2)/(448*c**4*d**5) + (b*d + 2*c*d*x)**(11/2)/(704*
c**4*d**7)

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Giac [A]  time = 1.16593, size = 252, normalized size = 2.08 \begin{align*} \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{64 \, \sqrt{2 \, c d x + b d} c^{4} d} + \frac{77 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} c^{40} d^{74} - 616 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b^{2} c^{41} d^{74} + 1232 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a^{2} c^{42} d^{74} - 33 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b^{2} c^{40} d^{72} + 132 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} a c^{41} d^{72} + 7 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}} c^{40} d^{70}}{4928 \, c^{44} d^{77}} \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)^(3/2),x, algorithm="giac")

[Out]

1/64*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/(sqrt(2*c*d*x + b*d)*c^4*d) + 1/4928*(77*(2*c*d*x + b*d)
^(3/2)*b^4*c^40*d^74 - 616*(2*c*d*x + b*d)^(3/2)*a*b^2*c^41*d^74 + 1232*(2*c*d*x + b*d)^(3/2)*a^2*c^42*d^74 -
33*(2*c*d*x + b*d)^(7/2)*b^2*c^40*d^72 + 132*(2*c*d*x + b*d)^(7/2)*a*c^41*d^72 + 7*(2*c*d*x + b*d)^(11/2)*c^40
*d^70)/(c^44*d^77)