3.1277 \(\int \sqrt{b d+2 c d x} (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=121 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac{(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]

[Out]

-((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(3/2))/(192*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(7/2))/(448*c^4*d^3)
- (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(11/2))/(704*c^4*d^5) + (b*d + 2*c*d*x)^(15/2)/(960*c^4*d^7)

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Rubi [A]  time = 0.0512324, 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} \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac{(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(3/2))/(192*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(7/2))/(448*c^4*d^3)
- (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(11/2))/(704*c^4*d^5) + (b*d + 2*c*d*x)^(15/2)/(960*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 \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3 \sqrt{b d+2 c d x}}{64 c^3}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{5/2}}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{9/2}}{64 c^3 d^4}+\frac{(b d+2 c d x)^{13/2}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac{(b d+2 c d x)^{15/2}}{960 c^4 d^7}\\ \end{align*}

Mathematica [A]  time = 0.0844362, size = 83, normalized size = 0.69 \[ \frac{\left (-315 \left (b^2-4 a c\right ) (b+2 c x)^4+495 \left (b^2-4 a c\right )^2 (b+2 c x)^2-385 \left (b^2-4 a c\right )^3+77 (b+2 c x)^6\right ) (d (b+2 c x))^{3/2}}{73920 c^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(b + 2*c*x))^(3/2)*(-385*(b^2 - 4*a*c)^3 + 495*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 315*(b^2 - 4*a*c)*(b + 2*c*
x)^4 + 77*(b + 2*c*x)^6))/(73920*c^4*d)

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Maple [A]  time = 0.046, 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}+315\,a{c}^{5}{x}^{4}+210\,{b}^{2}{c}^{4}{x}^{4}+630\,ab{c}^{4}{x}^{3}+35\,{b}^{3}{c}^{3}{x}^{3}+495\,{a}^{2}{c}^{4}{x}^{2}+225\,a{b}^{2}{c}^{3}{x}^{2}-15\,{b}^{4}{c}^{2}{x}^{2}+495\,{a}^{2}b{c}^{3}x-90\,a{b}^{3}{c}^{2}x+6\,{b}^{5}cx+385\,{a}^{3}{c}^{3}-165\,{a}^{2}{b}^{2}{c}^{2}+30\,a{b}^{4}c-2\,{b}^{6} \right ) }{1155\,{c}^{4}}\sqrt{2\,cdx+bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^3,x)

[Out]

1/1155*(2*c*x+b)*(77*c^6*x^6+231*b*c^5*x^5+315*a*c^5*x^4+210*b^2*c^4*x^4+630*a*b*c^4*x^3+35*b^3*c^3*x^3+495*a^
2*c^4*x^2+225*a*b^2*c^3*x^2-15*b^4*c^2*x^2+495*a^2*b*c^3*x-90*a*b^3*c^2*x+6*b^5*c*x+385*a^3*c^3-165*a^2*b^2*c^
2+30*a*b^4*c-2*b^6)*(2*c*d*x+b*d)^(1/2)/c^4

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Maxima [A]  time = 1.0426, size = 171, normalized size = 1.41 \begin{align*} -\frac{315 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 495 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} d^{4} + 385 \,{\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}} d^{6} - 77 \,{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}{73920 \, c^{4} d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

-1/73920*(315*(2*c*d*x + b*d)^(11/2)*(b^2 - 4*a*c)*d^2 - 495*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^(7
/2)*d^4 + 385*(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)*d^6 - 77*(2*c*d*x + b*d)^
(15/2))/(c^4*d^7)

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Fricas [A]  time = 2.03903, size = 462, normalized size = 3.82 \begin{align*} \frac{{\left (154 \, c^{7} x^{7} + 539 \, b c^{6} x^{6} - 2 \, b^{7} + 30 \, a b^{5} c - 165 \, a^{2} b^{3} c^{2} + 385 \, a^{3} b c^{3} + 21 \,{\left (31 \, b^{2} c^{5} + 30 \, a c^{6}\right )} x^{5} + 35 \,{\left (8 \, b^{3} c^{4} + 45 \, a b c^{5}\right )} x^{4} + 5 \,{\left (b^{4} c^{3} + 216 \, a b^{2} c^{4} + 198 \, a^{2} c^{5}\right )} x^{3} - 3 \,{\left (b^{5} c^{2} - 15 \, a b^{3} c^{3} - 495 \, a^{2} b c^{4}\right )} x^{2} +{\left (2 \, b^{6} c - 30 \, a b^{4} c^{2} + 165 \, a^{2} b^{2} c^{3} + 770 \, a^{3} c^{4}\right )} x\right )} \sqrt{2 \, c d x + b d}}{1155 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/1155*(154*c^7*x^7 + 539*b*c^6*x^6 - 2*b^7 + 30*a*b^5*c - 165*a^2*b^3*c^2 + 385*a^3*b*c^3 + 21*(31*b^2*c^5 +
30*a*c^6)*x^5 + 35*(8*b^3*c^4 + 45*a*b*c^5)*x^4 + 5*(b^4*c^3 + 216*a*b^2*c^4 + 198*a^2*c^5)*x^3 - 3*(b^5*c^2 -
15*a*b^3*c^3 - 495*a^2*b*c^4)*x^2 + (2*b^6*c - 30*a*b^4*c^2 + 165*a^2*b^2*c^3 + 770*a^3*c^4)*x)*sqrt(2*c*d*x
+ b*d)/c^4

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Sympy [A]  time = 4.47442, size = 151, normalized size = 1.25 \begin{align*} \frac{\frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}{192 c^{3}} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{448 c^{3} d^{2}} + \frac{\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{11}{2}}}{704 c^{3} d^{4}} + \frac{\left (b d + 2 c d x\right )^{\frac{15}{2}}}{960 c^{3} d^{6}}}{c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**3,x)

[Out]

((b*d + 2*c*d*x)**(3/2)*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)/(192*c**3) + (b*d + 2*c*d*x)**
(7/2)*(48*a**2*c**2 - 24*a*b**2*c + 3*b**4)/(448*c**3*d**2) + (12*a*c - 3*b**2)*(b*d + 2*c*d*x)**(11/2)/(704*c
**3*d**4) + (b*d + 2*c*d*x)**(15/2)/(960*c**3*d**6))/(c*d)

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Giac [B]  time = 1.18686, size = 1049, normalized size = 8.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/2882880*(960960*(2*c*d*x + b*d)^(3/2)*a^3 - 288288*(5*(2*c*d*x + b*d)^(3/2)*b*d - 3*(2*c*d*x + b*d)^(5/2))*a
^2*b/(c*d) + 20592*(35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 - 42*(2*c*d*x + b*d)^(5/2)*b*d + 15*(2*c*d*x + b*d)^(7/2)
)*a*b^2/(c^2*d^2) + 20592*(35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 - 42*(2*c*d*x + b*d)^(5/2)*b*d + 15*(2*c*d*x + b*d
)^(7/2))*a^2/(c*d^2) - 1144*(105*(2*c*d*x + b*d)^(3/2)*b^3*d^3 - 189*(2*c*d*x + b*d)^(5/2)*b^2*d^2 + 135*(2*c*
d*x + b*d)^(7/2)*b*d - 35*(2*c*d*x + b*d)^(9/2))*b^3/(c^3*d^3) - 6864*(105*(2*c*d*x + b*d)^(3/2)*b^3*d^3 - 189
*(2*c*d*x + b*d)^(5/2)*b^2*d^2 + 135*(2*c*d*x + b*d)^(7/2)*b*d - 35*(2*c*d*x + b*d)^(9/2))*a*b/(c^2*d^3) + 156
*(1155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 - 2772*(2*c*d*x + b*d)^(5/2)*b^3*d^3 + 2970*(2*c*d*x + b*d)^(7/2)*b^2*d^2
- 1540*(2*c*d*x + b*d)^(9/2)*b*d + 315*(2*c*d*x + b*d)^(11/2))*b^2/(c^3*d^4) + 156*(1155*(2*c*d*x + b*d)^(3/2
)*b^4*d^4 - 2772*(2*c*d*x + b*d)^(5/2)*b^3*d^3 + 2970*(2*c*d*x + b*d)^(7/2)*b^2*d^2 - 1540*(2*c*d*x + b*d)^(9/
2)*b*d + 315*(2*c*d*x + b*d)^(11/2))*a/(c^2*d^4) - 30*(3003*(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 + 12870*(2*c*d*x + b*d)^(7/2)*b^3*d^3 - 10010*(2*c*d*x + b*d)^(9/2)*b^2*d^2 + 4095*(2*c*d*x +
b*d)^(11/2)*b*d - 693*(2*c*d*x + b*d)^(13/2))*b/(c^3*d^5) + (15015*(2*c*d*x + b*d)^(3/2)*b^6*d^6 - 54054*(2*c
*d*x + b*d)^(5/2)*b^5*d^5 + 96525*(2*c*d*x + b*d)^(7/2)*b^4*d^4 - 100100*(2*c*d*x + b*d)^(9/2)*b^3*d^3 + 61425
*(2*c*d*x + b*d)^(11/2)*b^2*d^2 - 20790*(2*c*d*x + b*d)^(13/2)*b*d + 3003*(2*c*d*x + b*d)^(15/2))/(c^3*d^6))/(
c*d)