3.1255 \(\int \frac{(c+d x)^2}{(a+b x)^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0331657, antiderivative size = 65, 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} \[ -\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{d^2}{2 b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*x)^5,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0187041, size = 56, normalized size = 0.86 \[ -\frac{a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )}{12 b^3 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + b*x)^5,x]

[Out]

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

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Maple [A]  time = 0.005, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{2\,d \left ( ad-bc \right ) }{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.962005, size = 132, normalized size = 2.03 \begin{align*} -\frac{6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*b^2*d^2*x^2 + 3*b^2*c^2 + 2*a*b*c*d + a^2*d^2 + 4*(2*b^2*c*d + a*b*d^2)*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6
*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

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Fricas [A]  time = 2.00984, size = 201, normalized size = 3.09 \begin{align*} -\frac{6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*b^2*d^2*x^2 + 3*b^2*c^2 + 2*a*b*c*d + a^2*d^2 + 4*(2*b^2*c*d + a*b*d^2)*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6
*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

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Sympy [A]  time = 0.938739, size = 104, normalized size = 1.6 \begin{align*} - \frac{a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (4 a b d^{2} + 8 b^{2} c d\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a**2*d**2 + 2*a*b*c*d + 3*b**2*c**2 + 6*b**2*d**2*x**2 + x*(4*a*b*d**2 + 8*b**2*c*d))/(12*a**4*b**3 + 48*a**
3*b**4*x + 72*a**2*b**5*x**2 + 48*a*b**6*x**3 + 12*b**7*x**4)

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Giac [A]  time = 1.06059, size = 130, normalized size = 2. \begin{align*} -\frac{\frac{3 \, c^{2}}{{\left (b x + a\right )}^{4}} + \frac{8 \, c d}{{\left (b x + a\right )}^{3} b} - \frac{6 \, a c d}{{\left (b x + a\right )}^{4} b} + \frac{6 \, d^{2}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{8 \, a d^{2}}{{\left (b x + a\right )}^{3} b^{2}} + \frac{3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{4} b^{2}}}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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