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

Optimal. Leaf size=38 \[ \frac {b (c+d x)^4}{4 d^2}-\frac {(c+d x)^3 (b c-a d)}{3 d^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 38, 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} \begin {gather*} \frac {b (c+d x)^4}{4 d^2}-\frac {(c+d x)^3 (b c-a d)}{3 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 47, normalized size = 1.24 \begin {gather*} \frac {1}{12} x \left (4 d x^2 (a d+2 b c)+6 c x (2 a d+b c)+12 a c^2+3 b d^2 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(12*a*c^2 + 6*c*(b*c + 2*a*d)*x + 4*d*(2*b*c + a*d)*x^2 + 3*b*d^2*x^3))/12

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IntegrateAlgebraic [B]  time = 0.09, size = 88, normalized size = 2.32 \begin {gather*} \frac {(a+b x)^4 \left (\frac {6 a^2 d^2}{(a+b x)^2}+\frac {6 b^2 c^2}{(a+b x)^2}+\frac {8 b c d}{a+b x}-\frac {12 a b c d}{(a+b x)^2}-\frac {8 a d^2}{a+b x}+3 d^2\right )}{12 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^4*(3*d^2 + (6*b^2*c^2)/(a + b*x)^2 - (12*a*b*c*d)/(a + b*x)^2 + (6*a^2*d^2)/(a + b*x)^2 + (8*b*c*d)
/(a + b*x) - (8*a*d^2)/(a + b*x)))/(12*b^3)

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fricas [A]  time = 0.39, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac {1}{3} \, {\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{2} + 2 \, a c d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 49, normalized size = 1.29 \begin {gather*} \frac {1}{4} \, b d^{2} x^{4} + \frac {2}{3} \, b c d x^{3} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{2} \, b c^{2} x^{2} + a c d x^{2} + a c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 55, normalized size = 1.45 \begin {gather*} \frac {b \,d^{2} x^{4}}{4}+a \,c^{2} x +\frac {\left (b c d +\left (a d +b c \right ) d \right ) x^{3}}{3}+\frac {\left (a c d +\left (a d +b c \right ) c \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.02, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac {1}{3} \, {\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{2} + 2 \, a c d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 47, normalized size = 1.24 \begin {gather*} x^2\,\left (\frac {b\,c^2}{2}+a\,d\,c\right )+x^3\,\left (\frac {a\,d^2}{3}+\frac {2\,b\,c\,d}{3}\right )+\frac {b\,d^2\,x^4}{4}+a\,c^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.13, size = 49, normalized size = 1.29 \begin {gather*} a c^{2} x + \frac {b d^{2} x^{4}}{4} + x^{3} \left (\frac {a d^{2}}{3} + \frac {2 b c d}{3}\right ) + x^{2} \left (a c d + \frac {b c^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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