### 3.1280 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^{5/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.049903, 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} $-\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{64 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{9/2}}{576 c^4 d^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0692922, size = 83, normalized size = 0.69 $\frac{-27 \left (b^2-4 a c\right ) (b+2 c x)^4+135 \left (b^2-4 a c\right )^2 (b+2 c x)^2+15 \left (b^2-4 a c\right )^3+5 (b+2 c x)^6}{2880 c^4 d (d (b+2 c x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.043, size = 173, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -5\,{c}^{6}{x}^{6}-15\,b{c}^{5}{x}^{5}-27\,a{c}^{5}{x}^{4}-12\,{b}^{2}{c}^{4}{x}^{4}-54\,ab{c}^{4}{x}^{3}+{b}^{3}{c}^{3}{x}^{3}-135\,{a}^{2}{c}^{4}{x}^{2}+27\,a{b}^{2}{c}^{3}{x}^{2}-3\,{b}^{4}{c}^{2}{x}^{2}-135\,{a}^{2}b{c}^{3}x+54\,a{b}^{3}{c}^{2}x-6\,{b}^{5}cx+15\,{a}^{3}{c}^{3}-45\,{a}^{2}{b}^{2}{c}^{2}+18\,a{b}^{4}c-2\,{b}^{6} \right ) }{45\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{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)^(5/2),x)

[Out]

-1/45*(2*c*x+b)*(-5*c^6*x^6-15*b*c^5*x^5-27*a*c^5*x^4-12*b^2*c^4*x^4-54*a*b*c^4*x^3+b^3*c^3*x^3-135*a^2*c^4*x^
2+27*a*b^2*c^3*x^2-3*b^4*c^2*x^2-135*a^2*b*c^3*x+54*a*b^3*c^2*x-6*b^5*c*x+15*a^3*c^3-45*a^2*b^2*c^2+18*a*b^4*c
-2*b^6)/c^4/(2*c*d*x+b*d)^(5/2)

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Maxima [A]  time = 1.04897, size = 184, normalized size = 1.52 \begin{align*} \frac{\frac{15 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{3}} - \frac{27 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 135 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{2 \, c d x + b d} d^{4} - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{c^{3} d^{6}}}{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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.10276, size = 405, normalized size = 3.35 \begin{align*} \frac{{\left (5 \, c^{6} x^{6} + 15 \, b c^{5} x^{5} + 2 \, b^{6} - 18 \, a b^{4} c + 45 \, a^{2} b^{2} c^{2} - 15 \, a^{3} c^{3} + 3 \,{\left (4 \, b^{2} c^{4} + 9 \, a c^{5}\right )} x^{4} -{\left (b^{3} c^{3} - 54 \, a b c^{4}\right )} x^{3} + 3 \,{\left (b^{4} c^{2} - 9 \, a b^{2} c^{3} + 45 \, a^{2} c^{4}\right )} x^{2} + 3 \,{\left (2 \, b^{5} c - 18 \, a b^{3} c^{2} + 45 \, a^{2} b c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{45 \,{\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\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)^(5/2),x, algorithm="fricas")

[Out]

1/45*(5*c^6*x^6 + 15*b*c^5*x^5 + 2*b^6 - 18*a*b^4*c + 45*a^2*b^2*c^2 - 15*a^3*c^3 + 3*(4*b^2*c^4 + 9*a*c^5)*x^
4 - (b^3*c^3 - 54*a*b*c^4)*x^3 + 3*(b^4*c^2 - 9*a*b^2*c^3 + 45*a^2*c^4)*x^2 + 3*(2*b^5*c - 18*a*b^3*c^2 + 45*a
^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(4*c^6*d^3*x^2 + 4*b*c^5*d^3*x + b^2*c^4*d^3)

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Sympy [A]  time = 95.0958, size = 128, normalized size = 1.06 \begin{align*} - \frac{\left (4 a c - b^{2}\right )^{3}}{192 c^{4} d \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{b d + 2 c d x} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{64 c^{4} d^{3}} + \frac{\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}}{320 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}}}{576 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)**(5/2),x)

[Out]

-(4*a*c - b**2)**3/(192*c**4*d*(b*d + 2*c*d*x)**(3/2)) + sqrt(b*d + 2*c*d*x)*(48*a**2*c**2 - 24*a*b**2*c + 3*b
**4)/(64*c**4*d**3) + (12*a*c - 3*b**2)*(b*d + 2*c*d*x)**(5/2)/(320*c**4*d**5) + (b*d + 2*c*d*x)**(9/2)/(576*c
**4*d**7)

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Giac [A]  time = 1.18633, 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}}{192 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{4} d} + \frac{135 \, \sqrt{2 \, c d x + b d} b^{4} c^{32} d^{60} - 1080 \, \sqrt{2 \, c d x + b d} a b^{2} c^{33} d^{60} + 2160 \, \sqrt{2 \, c d x + b d} a^{2} c^{34} d^{60} - 27 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} c^{32} d^{58} + 108 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} a c^{33} d^{58} + 5 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{32} d^{56}}{2880 \, c^{36} d^{63}} \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)^(5/2),x, algorithm="giac")

[Out]

1/192*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/((2*c*d*x + b*d)^(3/2)*c^4*d) + 1/2880*(135*sqrt(2*c*d*
x + b*d)*b^4*c^32*d^60 - 1080*sqrt(2*c*d*x + b*d)*a*b^2*c^33*d^60 + 2160*sqrt(2*c*d*x + b*d)*a^2*c^34*d^60 - 2
7*(2*c*d*x + b*d)^(5/2)*b^2*c^32*d^58 + 108*(2*c*d*x + b*d)^(5/2)*a*c^33*d^58 + 5*(2*c*d*x + b*d)^(9/2)*c^32*d
^56)/(c^36*d^63)