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

Optimal. Leaf size=50 \[ \frac {(b c-a d)^2 \log (c+d x)}{d^3}-\frac {b x (b c-a d)}{d^2}+\frac {(a+b x)^2}{2 d} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 50, 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 x (b c-a d)}{d^2}+\frac {(b c-a d)^2 \log (c+d x)}{d^3}+\frac {(a+b x)^2}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 43, normalized size = 0.86 \begin {gather*} \frac {b d x (4 a d-2 b c+b d x)+2 (b c-a d)^2 \log (c+d x)}{2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^3}{a c+(b c+a d) x+b d x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 62, normalized size = 1.24 \begin {gather*} \frac {b^{2} d^{2} x^{2} - 2 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 60, normalized size = 1.20 \begin {gather*} \frac {b^{2} d x^{2} - 2 \, b^{2} c x + 4 \, a b d x}{2 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 74, normalized size = 1.48 \begin {gather*} \frac {b^{2} x^{2}}{2 d}+\frac {a^{2} \ln \left (d x +c \right )}{d}-\frac {2 a b c \ln \left (d x +c \right )}{d^{2}}+\frac {2 a b x}{d}+\frac {b^{2} c^{2} \ln \left (d x +c \right )}{d^{3}}-\frac {b^{2} c x}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.98, size = 60, normalized size = 1.20 \begin {gather*} \frac {b^{2} d x^{2} - 2 \, {\left (b^{2} c - 2 \, a b d\right )} x}{2 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.60, size = 62, normalized size = 1.24 \begin {gather*} \frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{d^3}-x\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )+\frac {b^2\,x^2}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 44, normalized size = 0.88 \begin {gather*} \frac {b^{2} x^{2}}{2 d} + x \left (\frac {2 a b}{d} - \frac {b^{2} c}{d^{2}}\right ) + \frac {\left (a d - b c\right )^{2} \log {\left (c + d x \right )}}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________