3.10.21 \(\int (b d+2 c d x)^4 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=45 \[ \frac {d^4 (b+2 c x)^7}{56 c^2}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5}{40 c^2} \]

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Rubi [A]  time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {683} \begin {gather*} \frac {d^4 (b+2 c x)^7}{56 c^2}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5}{40 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2),x]

[Out]

-((b^2 - 4*a*c)*d^4*(b + 2*c*x)^5)/(40*c^2) + (d^4*(b + 2*c*x)^7)/(56*c^2)

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)^4 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{4 c}+\frac {(b d+2 c d x)^6}{4 c d^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5}{40 c^2}+\frac {d^4 (b+2 c x)^7}{56 c^2}\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 102, normalized size = 2.27 \begin {gather*} d^4 \left (a b^4 x+\frac {8}{5} c^3 x^5 \left (2 a c+7 b^2\right )+8 b c^2 x^4 \left (a c+b^2\right )+b^2 c x^3 \left (8 a c+3 b^2\right )+\frac {1}{2} b^3 x^2 \left (8 a c+b^2\right )+8 b c^4 x^6+\frac {16 c^5 x^7}{7}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2),x]

[Out]

d^4*(a*b^4*x + (b^3*(b^2 + 8*a*c)*x^2)/2 + b^2*c*(3*b^2 + 8*a*c)*x^3 + 8*b*c^2*(b^2 + a*c)*x^4 + (8*c^3*(7*b^2
+ 2*a*c)*x^5)/5 + 8*b*c^4*x^6 + (16*c^5*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)^4 \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2), x]

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fricas [B]  time = 0.35, size = 137, normalized size = 3.04 \begin {gather*} \frac {16}{7} x^{7} d^{4} c^{5} + 8 x^{6} d^{4} c^{4} b + \frac {56}{5} x^{5} d^{4} c^{3} b^{2} + \frac {16}{5} x^{5} d^{4} c^{4} a + 8 x^{4} d^{4} c^{2} b^{3} + 8 x^{4} d^{4} c^{3} b a + 3 x^{3} d^{4} c b^{4} + 8 x^{3} d^{4} c^{2} b^{2} a + \frac {1}{2} x^{2} d^{4} b^{5} + 4 x^{2} d^{4} c b^{3} a + x d^{4} b^{4} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

16/7*x^7*d^4*c^5 + 8*x^6*d^4*c^4*b + 56/5*x^5*d^4*c^3*b^2 + 16/5*x^5*d^4*c^4*a + 8*x^4*d^4*c^2*b^3 + 8*x^4*d^4
*c^3*b*a + 3*x^3*d^4*c*b^4 + 8*x^3*d^4*c^2*b^2*a + 1/2*x^2*d^4*b^5 + 4*x^2*d^4*c*b^3*a + x*d^4*b^4*a

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giac [B]  time = 0.17, size = 137, normalized size = 3.04 \begin {gather*} \frac {16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + \frac {56}{5} \, b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a c^{4} d^{4} x^{5} + 8 \, b^{3} c^{2} d^{4} x^{4} + 8 \, a b c^{3} d^{4} x^{4} + 3 \, b^{4} c d^{4} x^{3} + 8 \, a b^{2} c^{2} d^{4} x^{3} + \frac {1}{2} \, b^{5} d^{4} x^{2} + 4 \, a b^{3} c d^{4} x^{2} + a b^{4} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

16/7*c^5*d^4*x^7 + 8*b*c^4*d^4*x^6 + 56/5*b^2*c^3*d^4*x^5 + 16/5*a*c^4*d^4*x^5 + 8*b^3*c^2*d^4*x^4 + 8*a*b*c^3
*d^4*x^4 + 3*b^4*c*d^4*x^3 + 8*a*b^2*c^2*d^4*x^3 + 1/2*b^5*d^4*x^2 + 4*a*b^3*c*d^4*x^2 + a*b^4*d^4*x

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maple [B]  time = 0.04, size = 137, normalized size = 3.04 \begin {gather*} \frac {16 c^{5} d^{4} x^{7}}{7}+8 b \,c^{4} d^{4} x^{6}+a \,b^{4} d^{4} x +\frac {\left (16 c^{4} d^{4} a +56 b^{2} d^{4} c^{3}\right ) x^{5}}{5}+\frac {\left (32 b \,c^{3} d^{4} a +32 b^{3} d^{4} c^{2}\right ) x^{4}}{4}+\frac {\left (24 b^{2} d^{4} c^{2} a +9 b^{4} d^{4} c \right ) x^{3}}{3}+\frac {\left (8 b^{3} d^{4} c a +b^{5} d^{4}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x)

[Out]

16/7*c^5*d^4*x^7+8*b*c^4*d^4*x^6+1/5*(16*a*c^4*d^4+56*b^2*c^3*d^4)*x^5+1/4*(32*a*b*c^3*d^4+32*b^3*c^2*d^4)*x^4
+1/3*(24*a*b^2*c^2*d^4+9*b^4*c*d^4)*x^3+1/2*(8*a*b^3*c*d^4+b^5*d^4)*x^2+b^4*d^4*a*x

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maxima [B]  time = 1.36, size = 120, normalized size = 2.67 \begin {gather*} \frac {16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + a b^{4} d^{4} x + \frac {8}{5} \, {\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{4} x^{5} + 8 \, {\left (b^{3} c^{2} + a b c^{3}\right )} d^{4} x^{4} + {\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (b^{5} + 8 \, a b^{3} c\right )} d^{4} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

16/7*c^5*d^4*x^7 + 8*b*c^4*d^4*x^6 + a*b^4*d^4*x + 8/5*(7*b^2*c^3 + 2*a*c^4)*d^4*x^5 + 8*(b^3*c^2 + a*b*c^3)*d
^4*x^4 + (3*b^4*c + 8*a*b^2*c^2)*d^4*x^3 + 1/2*(b^5 + 8*a*b^3*c)*d^4*x^2

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mupad [B]  time = 0.44, size = 113, normalized size = 2.51 \begin {gather*} \frac {16\,c^5\,d^4\,x^7}{7}+\frac {b^3\,d^4\,x^2\,\left (b^2+8\,a\,c\right )}{2}+8\,b\,c^4\,d^4\,x^6+\frac {8\,c^3\,d^4\,x^5\,\left (7\,b^2+2\,a\,c\right )}{5}+a\,b^4\,d^4\,x+8\,b\,c^2\,d^4\,x^4\,\left (b^2+a\,c\right )+b^2\,c\,d^4\,x^3\,\left (3\,b^2+8\,a\,c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2),x)

[Out]

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

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sympy [B]  time = 0.10, size = 143, normalized size = 3.18 \begin {gather*} a b^{4} d^{4} x + 8 b c^{4} d^{4} x^{6} + \frac {16 c^{5} d^{4} x^{7}}{7} + x^{5} \left (\frac {16 a c^{4} d^{4}}{5} + \frac {56 b^{2} c^{3} d^{4}}{5}\right ) + x^{4} \left (8 a b c^{3} d^{4} + 8 b^{3} c^{2} d^{4}\right ) + x^{3} \left (8 a b^{2} c^{2} d^{4} + 3 b^{4} c d^{4}\right ) + x^{2} \left (4 a b^{3} c d^{4} + \frac {b^{5} d^{4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a),x)

[Out]

a*b**4*d**4*x + 8*b*c**4*d**4*x**6 + 16*c**5*d**4*x**7/7 + x**5*(16*a*c**4*d**4/5 + 56*b**2*c**3*d**4/5) + x**
4*(8*a*b*c**3*d**4 + 8*b**3*c**2*d**4) + x**3*(8*a*b**2*c**2*d**4 + 3*b**4*c*d**4) + x**2*(4*a*b**3*c*d**4 + b
**5*d**4/2)

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