3.1787 \(\int \frac {(a c+(b c+a d) x+b d x^2)^3}{a+b x} \, dx\)

Optimal. Leaf size=65 \[ -\frac {2 b (c+d x)^5 (b c-a d)}{5 d^3}+\frac {(c+d x)^4 (b c-a d)^2}{4 d^3}+\frac {b^2 (c+d x)^6}{6 d^3} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac {2 b (c+d x)^5 (b c-a d)}{5 d^3}+\frac {(c+d x)^4 (b c-a d)^2}{4 d^3}+\frac {b^2 (c+d x)^6}{6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 122, normalized size = 1.88 \[ \frac {1}{4} d x^4 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{2} a c^2 x^2 (3 a d+2 b c)+\frac {1}{5} b d^2 x^5 (2 a d+3 b c)+\frac {1}{6} b^2 d^3 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.99, size = 124, normalized size = 1.91 \[ \frac {1}{6} \, b^{2} d^{3} x^{6} + a^{2} c^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 130, normalized size = 2.00 \[ \frac {1}{6} \, b^{2} d^{3} x^{6} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {2}{5} \, a b d^{3} x^{5} + \frac {3}{4} \, b^{2} c^{2} d x^{4} + \frac {3}{2} \, a b c d^{2} x^{4} + \frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{2} d x^{3} + a^{2} c d^{2} x^{3} + a b c^{3} x^{2} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*d^3*x^6 + 3/5*b^2*c*d^2*x^5 + 2/5*a*b*d^3*x^5 + 3/4*b^2*c^2*d*x^4 + 3/2*a*b*c*d^2*x^4 + 1/4*a^2*d^3*x^
4 + 1/3*b^2*c^3*x^3 + 2*a*b*c^2*d*x^3 + a^2*c*d^2*x^3 + a*b*c^3*x^2 + 3/2*a^2*c^2*d*x^2 + a^2*c^3*x

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maple [B]  time = 0.05, size = 147, normalized size = 2.26 \[ \frac {b^{2} d^{3} x^{6}}{6}+a^{2} c^{3} x +\frac {\left (b^{2} c \,d^{2}+2 \left (a d +b c \right ) b \,d^{2}\right ) x^{5}}{5}+\frac {\left (2 \left (a d +b c \right ) b c d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) d \right ) x^{4}}{4}+\frac {\left (2 \left (a d +b c \right ) a c d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) c \right ) x^{3}}{3}+\frac {\left (a^{2} c^{2} d +2 \left (a d +b c \right ) a \,c^{2}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.03, size = 124, normalized size = 1.91 \[ \frac {1}{6} \, b^{2} d^{3} x^{6} + a^{2} c^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.59, size = 115, normalized size = 1.77 \[ x^3\,\left (a^2\,c\,d^2+2\,a\,b\,c^2\,d+\frac {b^2\,c^3}{3}\right )+x^4\,\left (\frac {a^2\,d^3}{4}+\frac {3\,a\,b\,c\,d^2}{2}+\frac {3\,b^2\,c^2\,d}{4}\right )+a^2\,c^3\,x+\frac {b^2\,d^3\,x^6}{6}+\frac {a\,c^2\,x^2\,\left (3\,a\,d+2\,b\,c\right )}{2}+\frac {b\,d^2\,x^5\,\left (2\,a\,d+3\,b\,c\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.17, size = 133, normalized size = 2.05 \[ a^{2} c^{3} x + \frac {b^{2} d^{3} x^{6}}{6} + x^{5} \left (\frac {2 a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + \frac {3 a b c d^{2}}{2} + \frac {3 b^{2} c^{2} d}{4}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + \frac {b^{2} c^{3}}{3}\right ) + x^{2} \left (\frac {3 a^{2} c^{2} d}{2} + a b c^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**3*x + b**2*d**3*x**6/6 + x**5*(2*a*b*d**3/5 + 3*b**2*c*d**2/5) + x**4*(a**2*d**3/4 + 3*a*b*c*d**2/2 +
3*b**2*c**2*d/4) + x**3*(a**2*c*d**2 + 2*a*b*c**2*d + b**2*c**3/3) + x**2*(3*a**2*c**2*d/2 + a*b*c**3)

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