3.1344 \(\int \frac{(a+b x)^4}{(c+d x)^2} \, dx\)

Optimal. Leaf size=104 \[ -\frac{2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac{6 b^2 x (b c-a d)^2}{d^4}-\frac{(b c-a d)^4}{d^5 (c+d x)}-\frac{4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac{b^4 (c+d x)^3}{3 d^5} \]

[Out]

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

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Rubi [A]  time = 0.100315, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac{6 b^2 x (b c-a d)^2}{d^4}-\frac{(b c-a d)^4}{d^5 (c+d x)}-\frac{4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac{b^4 (c+d x)^3}{3 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(c + d*x)^2,x]

[Out]

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

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])

Rubi steps

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

Mathematica [A]  time = 0.0564887, size = 165, normalized size = 1.59 \[ \frac{18 a^2 b^2 d^2 \left (-c^2+c d x+d^2 x^2\right )+12 a^3 b c d^3-3 a^4 d^4+6 a b^3 d \left (-4 c^2 d x+2 c^3-3 c d^2 x^2+d^3 x^3\right )-12 b (c+d x) (b c-a d)^3 \log (c+d x)+b^4 \left (6 c^2 d^2 x^2+9 c^3 d x-3 c^4-2 c d^3 x^3+d^4 x^4\right )}{3 d^5 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x)^2,x]

[Out]

(12*a^3*b*c*d^3 - 3*a^4*d^4 + 18*a^2*b^2*d^2*(-c^2 + c*d*x + d^2*x^2) + 6*a*b^3*d*(2*c^3 - 4*c^2*d*x - 3*c*d^2
*x^2 + d^3*x^3) + b^4*(-3*c^4 + 9*c^3*d*x + 6*c^2*d^2*x^2 - 2*c*d^3*x^3 + d^4*x^4) - 12*b*(b*c - a*d)^3*(c + d
*x)*Log[c + d*x])/(3*d^5*(c + d*x))

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Maple [B]  time = 0.009, size = 230, normalized size = 2.2 \begin{align*}{\frac{{b}^{4}{x}^{3}}{3\,{d}^{2}}}+2\,{\frac{{b}^{3}{x}^{2}a}{{d}^{2}}}-{\frac{{b}^{4}{x}^{2}c}{{d}^{3}}}+6\,{\frac{{b}^{2}{a}^{2}x}{{d}^{2}}}-8\,{\frac{a{b}^{3}cx}{{d}^{3}}}+3\,{\frac{{b}^{4}{c}^{2}x}{{d}^{4}}}-{\frac{{a}^{4}}{d \left ( dx+c \right ) }}+4\,{\frac{{a}^{3}bc}{{d}^{2} \left ( dx+c \right ) }}-6\,{\frac{{b}^{2}{a}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+4\,{\frac{a{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) }}-{\frac{{b}^{4}{c}^{4}}{{d}^{5} \left ( dx+c \right ) }}+4\,{\frac{b\ln \left ( dx+c \right ){a}^{3}}{{d}^{2}}}-12\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{2}c}{{d}^{3}}}+12\,{\frac{{b}^{3}\ln \left ( dx+c \right ) a{c}^{2}}{{d}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( dx+c \right ){c}^{3}}{{d}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^4/d^2*x^3+2*b^3/d^2*x^2*a-b^4/d^3*x^2*c+6*b^2/d^2*a^2*x-8*b^3/d^3*a*c*x+3*b^4/d^4*c^2*x-1/d/(d*x+c)*a^4+
4/d^2/(d*x+c)*a^3*b*c-6/d^3/(d*x+c)*a^2*b^2*c^2+4/d^4/(d*x+c)*a*b^3*c^3-1/d^5/(d*x+c)*b^4*c^4+4*b/d^2*ln(d*x+c
)*a^3-12*b^2/d^3*ln(d*x+c)*a^2*c+12*b^3/d^4*ln(d*x+c)*a*c^2-4*b^4/d^5*ln(d*x+c)*c^3

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Maxima [A]  time = 0.955658, size = 247, normalized size = 2.38 \begin{align*} -\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{d^{6} x + c d^{5}} + \frac{b^{4} d^{2} x^{3} - 3 \,{\left (b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{2} + 3 \,{\left (3 \, b^{4} c^{2} - 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x}{3 \, d^{4}} - \frac{4 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (d x + c\right )}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.72844, size = 540, normalized size = 5.19 \begin{align*} \frac{b^{4} d^{4} x^{4} - 3 \, b^{4} c^{4} + 12 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} - 2 \,{\left (b^{4} c d^{3} - 3 \, a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 3 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 3 \,{\left (3 \, b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3}\right )} x - 12 \,{\left (b^{4} c^{4} - 3 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x\right )} \log \left (d x + c\right )}{3 \,{\left (d^{6} x + c d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^4*d^4*x^4 - 3*b^4*c^4 + 12*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 3*a^4*d^4 - 2*(b^4*c*d^3
- 3*a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 3*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x^2 + 3*(3*b^4*c^3*d - 8*a*b^3*c^2*d^2 +
6*a^2*b^2*c*d^3)*x - 12*(b^4*c^4 - 3*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 - a^3*b*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2
*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x)*log(d*x + c))/(d^6*x + c*d^5)

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Sympy [A]  time = 0.902879, size = 151, normalized size = 1.45 \begin{align*} \frac{b^{4} x^{3}}{3 d^{2}} + \frac{4 b \left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{5}} - \frac{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}}{c d^{5} + d^{6} x} + \frac{x^{2} \left (2 a b^{3} d - b^{4} c\right )}{d^{3}} + \frac{x \left (6 a^{2} b^{2} d^{2} - 8 a b^{3} c d + 3 b^{4} c^{2}\right )}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.0705, size = 331, normalized size = 3.18 \begin{align*} \frac{{\left (b^{4} - \frac{6 \,{\left (b^{4} c d - a b^{3} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac{18 \,{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}}\right )}{\left (d x + c\right )}^{3}}{3 \, d^{5}} + \frac{4 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d^{5}} - \frac{\frac{b^{4} c^{4} d^{3}}{d x + c} - \frac{4 \, a b^{3} c^{3} d^{4}}{d x + c} + \frac{6 \, a^{2} b^{2} c^{2} d^{5}}{d x + c} - \frac{4 \, a^{3} b c d^{6}}{d x + c} + \frac{a^{4} d^{7}}{d x + c}}{d^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^4 - 6*(b^4*c*d - a*b^3*d^2)/((d*x + c)*d) + 18*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)/((d*x + c)^2
*d^2))*(d*x + c)^3/d^5 + 4*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*log(abs(d*x + c)/((d*x + c)
^2*abs(d)))/d^5 - (b^4*c^4*d^3/(d*x + c) - 4*a*b^3*c^3*d^4/(d*x + c) + 6*a^2*b^2*c^2*d^5/(d*x + c) - 4*a^3*b*c
*d^6/(d*x + c) + a^4*d^7/(d*x + c))/d^8