3.10.36 \(\int (b d+2 c d x)^2 (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=73 \[ -\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac {d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3} \]

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Rubi [A]  time = 0.08, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \begin {gather*} -\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac {d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3)/(96*c^3) - ((b^2 - 4*a*c)*d^2*(b + 2*c*x)^5)/(80*c^3) + (d^2*(b + 2*c*x)^7
)/(224*c^3)

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 (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^2}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{8 c^2 d^2}+\frac {(b d+2 c d x)^6}{16 c^2 d^4}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3}{96 c^3}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^5}{80 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 111, normalized size = 1.52 \begin {gather*} d^2 \left (a^2 b^2 x+\frac {1}{3} x^3 \left (4 a^2 c^2+10 a b^2 c+b^4\right )+\frac {1}{5} c^2 x^5 \left (8 a c+13 b^2\right )+\frac {1}{2} b c x^4 \left (8 a c+3 b^2\right )+a b x^2 \left (2 a c+b^2\right )+2 b c^3 x^6+\frac {4 c^4 x^7}{7}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

d^2*(a^2*b^2*x + a*b*(b^2 + 2*a*c)*x^2 + ((b^4 + 10*a*b^2*c + 4*a^2*c^2)*x^3)/3 + (b*c*(3*b^2 + 8*a*c)*x^4)/2
+ (c^2*(13*b^2 + 8*a*c)*x^5)/5 + 2*b*c^3*x^6 + (4*c^4*x^7)/7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^2, x]

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fricas [B]  time = 0.35, size = 148, normalized size = 2.03 \begin {gather*} \frac {4}{7} x^{7} d^{2} c^{4} + 2 x^{6} d^{2} c^{3} b + \frac {13}{5} x^{5} d^{2} c^{2} b^{2} + \frac {8}{5} x^{5} d^{2} c^{3} a + \frac {3}{2} x^{4} d^{2} c b^{3} + 4 x^{4} d^{2} c^{2} b a + \frac {1}{3} x^{3} d^{2} b^{4} + \frac {10}{3} x^{3} d^{2} c b^{2} a + \frac {4}{3} x^{3} d^{2} c^{2} a^{2} + x^{2} d^{2} b^{3} a + 2 x^{2} d^{2} c b a^{2} + x d^{2} b^{2} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*x^7*d^2*c^4 + 2*x^6*d^2*c^3*b + 13/5*x^5*d^2*c^2*b^2 + 8/5*x^5*d^2*c^3*a + 3/2*x^4*d^2*c*b^3 + 4*x^4*d^2*c
^2*b*a + 1/3*x^3*d^2*b^4 + 10/3*x^3*d^2*c*b^2*a + 4/3*x^3*d^2*c^2*a^2 + x^2*d^2*b^3*a + 2*x^2*d^2*c*b*a^2 + x*
d^2*b^2*a^2

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giac [B]  time = 0.15, size = 148, normalized size = 2.03 \begin {gather*} \frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {13}{5} \, b^{2} c^{2} d^{2} x^{5} + \frac {8}{5} \, a c^{3} d^{2} x^{5} + \frac {3}{2} \, b^{3} c d^{2} x^{4} + 4 \, a b c^{2} d^{2} x^{4} + \frac {1}{3} \, b^{4} d^{2} x^{3} + \frac {10}{3} \, a b^{2} c d^{2} x^{3} + \frac {4}{3} \, a^{2} c^{2} d^{2} x^{3} + a b^{3} d^{2} x^{2} + 2 \, a^{2} b c d^{2} x^{2} + a^{2} b^{2} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 13/5*b^2*c^2*d^2*x^5 + 8/5*a*c^3*d^2*x^5 + 3/2*b^3*c*d^2*x^4 + 4*a*b*c^2*d
^2*x^4 + 1/3*b^4*d^2*x^3 + 10/3*a*b^2*c*d^2*x^3 + 4/3*a^2*c^2*d^2*x^3 + a*b^3*d^2*x^2 + 2*a^2*b*c*d^2*x^2 + a^
2*b^2*d^2*x

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maple [B]  time = 0.04, size = 176, normalized size = 2.41 \begin {gather*} \frac {4 c^{4} d^{2} x^{7}}{7}+2 b \,c^{3} d^{2} x^{6}+a^{2} b^{2} d^{2} x +\frac {\left (9 b^{2} c^{2} d^{2}+4 \left (2 a c +b^{2}\right ) c^{2} d^{2}\right ) x^{5}}{5}+\frac {\left (8 a b \,c^{2} d^{2}+2 b^{3} c \,d^{2}+4 \left (2 a c +b^{2}\right ) b c \,d^{2}\right ) x^{4}}{4}+\frac {\left (4 a^{2} c^{2} d^{2}+8 a \,b^{2} c \,d^{2}+\left (2 a c +b^{2}\right ) b^{2} d^{2}\right ) x^{3}}{3}+\frac {\left (4 a^{2} b c \,d^{2}+2 b^{3} d^{2} a \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*c^4*d^2*x^7+2*b*c^3*d^2*x^6+1/5*(9*b^2*d^2*c^2+4*c^2*d^2*(2*a*c+b^2))*x^5+1/4*(2*b^3*d^2*c+4*b*c*d^2*(2*a*
c+b^2)+8*c^2*d^2*a*b)*x^4+1/3*(b^2*d^2*(2*a*c+b^2)+8*b^2*c*d^2*a+4*c^2*d^2*a^2)*x^3+1/2*(4*a^2*b*c*d^2+2*a*b^3
*d^2)*x^2+b^2*d^2*a^2*x

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maxima [A]  time = 1.36, size = 127, normalized size = 1.74 \begin {gather*} \frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {1}{5} \, {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac {1}{2} \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} + {\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 1/5*(13*b^2*c^2 + 8*a*c^3)*d^2*x^5 + a^2*b^2*d^2*x + 1/2*(3*b^3*c + 8*a*b*
c^2)*d^2*x^4 + 1/3*(b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*x^3 + (a*b^3 + 2*a^2*b*c)*d^2*x^2

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mupad [B]  time = 0.43, size = 120, normalized size = 1.64 \begin {gather*} \frac {4\,c^4\,d^2\,x^7}{7}+\frac {d^2\,x^3\,\left (4\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{3}+a^2\,b^2\,d^2\,x+2\,b\,c^3\,d^2\,x^6+\frac {c^2\,d^2\,x^5\,\left (13\,b^2+8\,a\,c\right )}{5}+a\,b\,d^2\,x^2\,\left (b^2+2\,a\,c\right )+\frac {b\,c\,d^2\,x^4\,\left (3\,b^2+8\,a\,c\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c^4*d^2*x^7)/7 + (d^2*x^3*(b^4 + 4*a^2*c^2 + 10*a*b^2*c))/3 + a^2*b^2*d^2*x + 2*b*c^3*d^2*x^6 + (c^2*d^2*x^
5*(8*a*c + 13*b^2))/5 + a*b*d^2*x^2*(2*a*c + b^2) + (b*c*d^2*x^4*(8*a*c + 3*b^2))/2

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sympy [B]  time = 0.10, size = 156, normalized size = 2.14 \begin {gather*} a^{2} b^{2} d^{2} x + 2 b c^{3} d^{2} x^{6} + \frac {4 c^{4} d^{2} x^{7}}{7} + x^{5} \left (\frac {8 a c^{3} d^{2}}{5} + \frac {13 b^{2} c^{2} d^{2}}{5}\right ) + x^{4} \left (4 a b c^{2} d^{2} + \frac {3 b^{3} c d^{2}}{2}\right ) + x^{3} \left (\frac {4 a^{2} c^{2} d^{2}}{3} + \frac {10 a b^{2} c d^{2}}{3} + \frac {b^{4} d^{2}}{3}\right ) + x^{2} \left (2 a^{2} b c d^{2} + a b^{3} d^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b**2*d**2*x + 2*b*c**3*d**2*x**6 + 4*c**4*d**2*x**7/7 + x**5*(8*a*c**3*d**2/5 + 13*b**2*c**2*d**2/5) + x*
*4*(4*a*b*c**2*d**2 + 3*b**3*c*d**2/2) + x**3*(4*a**2*c**2*d**2/3 + 10*a*b**2*c*d**2/3 + b**4*d**2/3) + x**2*(
2*a**2*b*c*d**2 + a*b**3*d**2)

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