3.10.48 \(\int (a+b x)^5 (a c+b c x)^2 \, dx\)

Optimal. Leaf size=17 \[ \frac {c^2 (a+b x)^8}{8 b} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \begin {gather*} \frac {c^2 (a+b x)^8}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(a*c + b*c*x)^2,x]

[Out]

(c^2*(a + b*x)^8)/(8*b)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int (a+b x)^5 (a c+b c x)^2 \, dx &=c^2 \int (a+b x)^7 \, dx\\ &=\frac {c^2 (a+b x)^8}{8 b}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (a+b x)^8}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(a + b*x)^8)/(8*b)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^5 (a c+b c x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5*(a*c + b*c*x)^2,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5*(a*c + b*c*x)^2, x]

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fricas [B]  time = 0.98, size = 99, normalized size = 5.82 \begin {gather*} \frac {1}{8} x^{8} c^{2} b^{7} + x^{7} c^{2} b^{6} a + \frac {7}{2} x^{6} c^{2} b^{5} a^{2} + 7 x^{5} c^{2} b^{4} a^{3} + \frac {35}{4} x^{4} c^{2} b^{3} a^{4} + 7 x^{3} c^{2} b^{2} a^{5} + \frac {7}{2} x^{2} c^{2} b a^{6} + x c^{2} a^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^2,x, algorithm="fricas")

[Out]

1/8*x^8*c^2*b^7 + x^7*c^2*b^6*a + 7/2*x^6*c^2*b^5*a^2 + 7*x^5*c^2*b^4*a^3 + 35/4*x^4*c^2*b^3*a^4 + 7*x^3*c^2*b
^2*a^5 + 7/2*x^2*c^2*b*a^6 + x*c^2*a^7

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giac [B]  time = 0.99, size = 99, normalized size = 5.82 \begin {gather*} \frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^2,x, algorithm="giac")

[Out]

1/8*b^7*c^2*x^8 + a*b^6*c^2*x^7 + 7/2*a^2*b^5*c^2*x^6 + 7*a^3*b^4*c^2*x^5 + 35/4*a^4*b^3*c^2*x^4 + 7*a^5*b^2*c
^2*x^3 + 7/2*a^6*b*c^2*x^2 + a^7*c^2*x

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maple [B]  time = 0.00, size = 100, normalized size = 5.88 \begin {gather*} \frac {1}{8} b^{7} c^{2} x^{8}+a \,b^{6} c^{2} x^{7}+\frac {7}{2} a^{2} b^{5} c^{2} x^{6}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35}{4} a^{4} b^{3} c^{2} x^{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7}{2} a^{6} b \,c^{2} x^{2}+a^{7} c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(b*c*x+a*c)^2,x)

[Out]

1/8*b^7*c^2*x^8+a*b^6*c^2*x^7+7/2*a^2*b^5*c^2*x^6+7*a^3*b^4*c^2*x^5+35/4*a^4*b^3*c^2*x^4+7*a^5*b^2*c^2*x^3+7/2
*a^6*c^2*b*x^2+a^7*c^2*x

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maxima [B]  time = 1.36, size = 99, normalized size = 5.82 \begin {gather*} \frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^2,x, algorithm="maxima")

[Out]

1/8*b^7*c^2*x^8 + a*b^6*c^2*x^7 + 7/2*a^2*b^5*c^2*x^6 + 7*a^3*b^4*c^2*x^5 + 35/4*a^4*b^3*c^2*x^4 + 7*a^5*b^2*c
^2*x^3 + 7/2*a^6*b*c^2*x^2 + a^7*c^2*x

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mupad [B]  time = 0.04, size = 99, normalized size = 5.82 \begin {gather*} a^7\,c^2\,x+\frac {7\,a^6\,b\,c^2\,x^2}{2}+7\,a^5\,b^2\,c^2\,x^3+\frac {35\,a^4\,b^3\,c^2\,x^4}{4}+7\,a^3\,b^4\,c^2\,x^5+\frac {7\,a^2\,b^5\,c^2\,x^6}{2}+a\,b^6\,c^2\,x^7+\frac {b^7\,c^2\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + b*c*x)^2*(a + b*x)^5,x)

[Out]

a^7*c^2*x + (b^7*c^2*x^8)/8 + (7*a^6*b*c^2*x^2)/2 + a*b^6*c^2*x^7 + 7*a^5*b^2*c^2*x^3 + (35*a^4*b^3*c^2*x^4)/4
+ 7*a^3*b^4*c^2*x^5 + (7*a^2*b^5*c^2*x^6)/2

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sympy [B]  time = 0.10, size = 110, normalized size = 6.47 \begin {gather*} a^{7} c^{2} x + \frac {7 a^{6} b c^{2} x^{2}}{2} + 7 a^{5} b^{2} c^{2} x^{3} + \frac {35 a^{4} b^{3} c^{2} x^{4}}{4} + 7 a^{3} b^{4} c^{2} x^{5} + \frac {7 a^{2} b^{5} c^{2} x^{6}}{2} + a b^{6} c^{2} x^{7} + \frac {b^{7} c^{2} x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(b*c*x+a*c)**2,x)

[Out]

a**7*c**2*x + 7*a**6*b*c**2*x**2/2 + 7*a**5*b**2*c**2*x**3 + 35*a**4*b**3*c**2*x**4/4 + 7*a**3*b**4*c**2*x**5
+ 7*a**2*b**5*c**2*x**6/2 + a*b**6*c**2*x**7 + b**7*c**2*x**8/8

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