### 3.1278 $$\int \frac{(a+b x+c x^2)^3}{\sqrt{b d+2 c d x}} \, dx$$

Optimal. Leaf size=121 $-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}{64 c^4 d}+\frac{(b d+2 c d x)^{13/2}}{832 c^4 d^7}$

[Out]

-((b^2 - 4*a*c)^3*Sqrt[b*d + 2*c*d*x])/(64*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(5/2))/(320*c^4*d^3) -
((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2))/(192*c^4*d^5) + (b*d + 2*c*d*x)^(13/2)/(832*c^4*d^7)

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Rubi [A]  time = 0.0499577, 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} $-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}{64 c^4 d}+\frac{(b d+2 c d x)^{13/2}}{832 c^4 d^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((b^2 - 4*a*c)^3*Sqrt[b*d + 2*c*d*x])/(64*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(5/2))/(320*c^4*d^3) -
((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2))/(192*c^4*d^5) + (b*d + 2*c*d*x)^(13/2)/(832*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}{\sqrt{b d+2 c d x}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 \sqrt{b d+2 c d x}}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{7/2}}{64 c^3 d^4}+\frac{(b d+2 c d x)^{11/2}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}{64 c^4 d}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac{(b d+2 c d x)^{13/2}}{832 c^4 d^7}\\ \end{align*}

Mathematica [A]  time = 0.0761271, size = 83, normalized size = 0.69 $\frac{\left (-65 \left (b^2-4 a c\right ) (b+2 c x)^4+117 \left (b^2-4 a c\right )^2 (b+2 c x)^2-195 \left (b^2-4 a c\right )^3+15 (b+2 c x)^6\right ) \sqrt{d (b+2 c x)}}{12480 c^4 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[d*(b + 2*c*x)]*(-195*(b^2 - 4*a*c)^3 + 117*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 65*(b^2 - 4*a*c)*(b + 2*c*x)^
4 + 15*(b + 2*c*x)^6))/(12480*c^4*d)

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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}+65\,a{c}^{5}{x}^{4}+40\,{b}^{2}{c}^{4}{x}^{4}+130\,ab{c}^{4}{x}^{3}+5\,{b}^{3}{c}^{3}{x}^{3}+117\,{a}^{2}{c}^{4}{x}^{2}+39\,a{b}^{2}{c}^{3}{x}^{2}-3\,{b}^{4}{c}^{2}{x}^{2}+117\,{a}^{2}b{c}^{3}x-26\,a{b}^{3}{c}^{2}x+2\,{b}^{5}cx+195\,{a}^{3}{c}^{3}-117\,{a}^{2}{b}^{2}{c}^{2}+26\,a{b}^{4}c-2\,{b}^{6} \right ) }{195\,{c}^{4}}{\frac{1}{\sqrt{2\,cdx+bd}}}} \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)^(1/2),x)

[Out]

1/195*(2*c*x+b)*(15*c^6*x^6+45*b*c^5*x^5+65*a*c^5*x^4+40*b^2*c^4*x^4+130*a*b*c^4*x^3+5*b^3*c^3*x^3+117*a^2*c^4
*x^2+39*a*b^2*c^3*x^2-3*b^4*c^2*x^2+117*a^2*b*c^3*x-26*a*b^3*c^2*x+2*b^5*c*x+195*a^3*c^3-117*a^2*b^2*c^2+26*a*
b^4*c-2*b^6)/c^4/(2*c*d*x+b*d)^(1/2)

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Maxima [B]  time = 1.12018, size = 1050, normalized size = 8.68 \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)^(1/2),x, algorithm="maxima")

[Out]

1/960960*(960960*sqrt(2*c*d*x + b*d)*a^3 - 48048*a^2*(10*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*b
/(c*d) - (15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))/(c*d^2)) +
572*a*(84*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*b^2/(c^2*d
^2) - 36*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5
*(2*c*d*x + b*d)^(7/2))*b/(c^2*d^3) + (315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 3
78*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))/(c^2*d^4)) - 3432
*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*
x + b*d)^(7/2))*b^3/(c^3*d^3) + 572*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378
*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*b^2/(c^3*d^4) - 130
*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5/2)*b^3*d^3 -
990*(2*c*d*x + b*d)^(7/2)*b^2*d^2 + 385*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b*d)^(11/2))*b/(c^3*d^5) + 5
*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 - 6006*(2*c*d*x + b*d)^(3/2)*b^5*d^5 + 9009*(2*c*d*x + b*d)^(5/2)*b^4*d^4 -
8580*(2*c*d*x + b*d)^(7/2)*b^3*d^3 + 5005*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d*x + b*d)^(11/2)*b*d + 2
31*(2*c*d*x + b*d)^(13/2))/(c^3*d^6))/(c*d)

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Fricas [A]  time = 2.02513, size = 363, normalized size = 3. \begin{align*} \frac{{\left (15 \, c^{6} x^{6} + 45 \, b c^{5} x^{5} - 2 \, b^{6} + 26 \, a b^{4} c - 117 \, a^{2} b^{2} c^{2} + 195 \, a^{3} c^{3} + 5 \,{\left (8 \, b^{2} c^{4} + 13 \, a c^{5}\right )} x^{4} + 5 \,{\left (b^{3} c^{3} + 26 \, a b c^{4}\right )} x^{3} - 3 \,{\left (b^{4} c^{2} - 13 \, a b^{2} c^{3} - 39 \, a^{2} c^{4}\right )} x^{2} +{\left (2 \, b^{5} c - 26 \, a b^{3} c^{2} + 117 \, a^{2} b c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{195 \, c^{4} 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)^(1/2),x, algorithm="fricas")

[Out]

1/195*(15*c^6*x^6 + 45*b*c^5*x^5 - 2*b^6 + 26*a*b^4*c - 117*a^2*b^2*c^2 + 195*a^3*c^3 + 5*(8*b^2*c^4 + 13*a*c^
5)*x^4 + 5*(b^3*c^3 + 26*a*b*c^4)*x^3 - 3*(b^4*c^2 - 13*a*b^2*c^3 - 39*a^2*c^4)*x^2 + (2*b^5*c - 26*a*b^3*c^2
+ 117*a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(c^4*d)

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Sympy [A]  time = 161.051, size = 1363, normalized size = 11.26 \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)**(1/2),x)

[Out]

Piecewise((-(a**3*b/sqrt(b*d + 2*c*d*x) + a**3*(-b*d/sqrt(b*d + 2*c*d*x) - sqrt(b*d + 2*c*d*x))/d + 3*a**2*b**
2*(-b*d/sqrt(b*d + 2*c*d*x) - sqrt(b*d + 2*c*d*x))/(2*c*d) + 9*a**2*b*(b**2*d**2/sqrt(b*d + 2*c*d*x) + 2*b*d*s
qrt(b*d + 2*c*d*x) - (b*d + 2*c*d*x)**(3/2)/3)/(4*c*d**2) + 3*a**2*(-b**3*d**3/sqrt(b*d + 2*c*d*x) - 3*b**2*d*
*2*sqrt(b*d + 2*c*d*x) + b*d*(b*d + 2*c*d*x)**(3/2) - (b*d + 2*c*d*x)**(5/2)/5)/(4*c*d**3) + 3*a*b**3*(b**2*d*
*2/sqrt(b*d + 2*c*d*x) + 2*b*d*sqrt(b*d + 2*c*d*x) - (b*d + 2*c*d*x)**(3/2)/3)/(4*c**2*d**2) + 3*a*b**2*(-b**3
*d**3/sqrt(b*d + 2*c*d*x) - 3*b**2*d**2*sqrt(b*d + 2*c*d*x) + b*d*(b*d + 2*c*d*x)**(3/2) - (b*d + 2*c*d*x)**(5
/2)/5)/(2*c**2*d**3) + 15*a*b*(b**4*d**4/sqrt(b*d + 2*c*d*x) + 4*b**3*d**3*sqrt(b*d + 2*c*d*x) - 2*b**2*d**2*(
b*d + 2*c*d*x)**(3/2) + 4*b*d*(b*d + 2*c*d*x)**(5/2)/5 - (b*d + 2*c*d*x)**(7/2)/7)/(16*c**2*d**4) + 3*a*(-b**5
*d**5/sqrt(b*d + 2*c*d*x) - 5*b**4*d**4*sqrt(b*d + 2*c*d*x) + 10*b**3*d**3*(b*d + 2*c*d*x)**(3/2)/3 - 2*b**2*d
**2*(b*d + 2*c*d*x)**(5/2) + 5*b*d*(b*d + 2*c*d*x)**(7/2)/7 - (b*d + 2*c*d*x)**(9/2)/9)/(16*c**2*d**5) + b**4*
(-b**3*d**3/sqrt(b*d + 2*c*d*x) - 3*b**2*d**2*sqrt(b*d + 2*c*d*x) + b*d*(b*d + 2*c*d*x)**(3/2) - (b*d + 2*c*d*
x)**(5/2)/5)/(8*c**3*d**3) + 5*b**3*(b**4*d**4/sqrt(b*d + 2*c*d*x) + 4*b**3*d**3*sqrt(b*d + 2*c*d*x) - 2*b**2*
d**2*(b*d + 2*c*d*x)**(3/2) + 4*b*d*(b*d + 2*c*d*x)**(5/2)/5 - (b*d + 2*c*d*x)**(7/2)/7)/(16*c**3*d**4) + 9*b*
*2*(-b**5*d**5/sqrt(b*d + 2*c*d*x) - 5*b**4*d**4*sqrt(b*d + 2*c*d*x) + 10*b**3*d**3*(b*d + 2*c*d*x)**(3/2)/3 -
2*b**2*d**2*(b*d + 2*c*d*x)**(5/2) + 5*b*d*(b*d + 2*c*d*x)**(7/2)/7 - (b*d + 2*c*d*x)**(9/2)/9)/(32*c**3*d**5
) + 7*b*(b**6*d**6/sqrt(b*d + 2*c*d*x) + 6*b**5*d**5*sqrt(b*d + 2*c*d*x) - 5*b**4*d**4*(b*d + 2*c*d*x)**(3/2)
+ 4*b**3*d**3*(b*d + 2*c*d*x)**(5/2) - 15*b**2*d**2*(b*d + 2*c*d*x)**(7/2)/7 + 2*b*d*(b*d + 2*c*d*x)**(9/2)/3
- (b*d + 2*c*d*x)**(11/2)/11)/(64*c**3*d**6) + (-b**7*d**7/sqrt(b*d + 2*c*d*x) - 7*b**6*d**6*sqrt(b*d + 2*c*d*
x) + 7*b**5*d**5*(b*d + 2*c*d*x)**(3/2) - 7*b**4*d**4*(b*d + 2*c*d*x)**(5/2) + 5*b**3*d**3*(b*d + 2*c*d*x)**(7
/2) - 7*b**2*d**2*(b*d + 2*c*d*x)**(9/2)/3 + 7*b*d*(b*d + 2*c*d*x)**(11/2)/11 - (b*d + 2*c*d*x)**(13/2)/13)/(6
4*c**3*d**7))/c, Ne(c, 0)), (Piecewise((a**3*x, Eq(b, 0)), ((a + b*x)**4/(4*b), True))/sqrt(b*d), True))

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Giac [B]  time = 1.17592, size = 1050, normalized size = 8.68 \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)^(1/2),x, algorithm="giac")

[Out]

1/960960*(960960*sqrt(2*c*d*x + b*d)*a^3 - 480480*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*a^2*b/(c
*d) + 48048*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a*b^2/(c
^2*d^2) + 48048*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a^2/
(c*d^2) - 3432*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b
*d - 5*(2*c*d*x + b*d)^(7/2))*b^3/(c^3*d^3) - 20592*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)
*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*a*b/(c^2*d^3) + 572*(315*sqrt(2*c*d*x + b*d
)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*
b*d + 35*(2*c*d*x + b*d)^(9/2))*b^2/(c^3*d^4) + 572*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/
2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*a/(
c^2*d^4) - 130*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5
/2)*b^3*d^3 - 990*(2*c*d*x + b*d)^(7/2)*b^2*d^2 + 385*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b*d)^(11/2))*b
/(c^3*d^5) + 5*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 - 6006*(2*c*d*x + b*d)^(3/2)*b^5*d^5 + 9009*(2*c*d*x + b*d)^(
5/2)*b^4*d^4 - 8580*(2*c*d*x + b*d)^(7/2)*b^3*d^3 + 5005*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d*x + b*d)^
(11/2)*b*d + 231*(2*c*d*x + b*d)^(13/2))/(c^3*d^6))/(c*d)