3.1257 \(\int \frac{(c+d x)^2}{(a+b x)^7} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0329904, 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} \[ -\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*x)^7,x]

[Out]

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

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)^7} \, dx &=\int \left (\frac{(b c-a d)^2}{b^2 (a+b x)^7}+\frac{2 d (b c-a d)}{b^2 (a+b x)^6}+\frac{d^2}{b^2 (a+b x)^5}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{d^2}{4 b^3 (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0202034, size = 58, normalized size = 0.89 \[ -\frac{a^2 d^2+2 a b d (2 c+3 d x)+b^2 \left (10 c^2+24 c d x+15 d^2 x^2\right )}{60 b^3 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + b*x)^7,x]

[Out]

-(a^2*d^2 + 2*a*b*d*(2*c + 3*d*x) + b^2*(10*c^2 + 24*c*d*x + 15*d^2*x^2))/(60*b^3*(a + b*x)^6)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2}}{6\,{b}^{3} \left ( bx+a \right ) ^{6}}}+{\frac{2\,d \left ( ad-bc \right ) }{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{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)^7,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.15195, size = 162, normalized size = 2.49 \begin{align*} -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/60*(15*b^2*d^2*x^2 + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2 + 6*(4*b^2*c*d + a*b*d^2)*x)/(b^9*x^6 + 6*a*b^8*x^5 +
15*a^2*b^7*x^4 + 20*a^3*b^6*x^3 + 15*a^4*b^5*x^2 + 6*a^5*b^4*x + a^6*b^3)

________________________________________________________________________________________

Fricas [B]  time = 1.91312, size = 251, normalized size = 3.86 \begin{align*} -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/60*(15*b^2*d^2*x^2 + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2 + 6*(4*b^2*c*d + a*b*d^2)*x)/(b^9*x^6 + 6*a*b^8*x^5 +
15*a^2*b^7*x^4 + 20*a^3*b^6*x^3 + 15*a^4*b^5*x^2 + 6*a^5*b^4*x + a^6*b^3)

________________________________________________________________________________________

Sympy [B]  time = 1.34307, size = 128, normalized size = 1.97 \begin{align*} - \frac{a^{2} d^{2} + 4 a b c d + 10 b^{2} c^{2} + 15 b^{2} d^{2} x^{2} + x \left (6 a b d^{2} + 24 b^{2} c d\right )}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(b*x+a)**7,x)

[Out]

-(a**2*d**2 + 4*a*b*c*d + 10*b**2*c**2 + 15*b**2*d**2*x**2 + x*(6*a*b*d**2 + 24*b**2*c*d))/(60*a**6*b**3 + 360
*a**5*b**4*x + 900*a**4*b**5*x**2 + 1200*a**3*b**6*x**3 + 900*a**2*b**7*x**4 + 360*a*b**8*x**5 + 60*b**9*x**6)

________________________________________________________________________________________

Giac [A]  time = 1.07213, size = 82, normalized size = 1.26 \begin{align*} -\frac{15 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 6 \, a b d^{2} x + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}}{60 \,{\left (b x + a\right )}^{6} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/60*(15*b^2*d^2*x^2 + 24*b^2*c*d*x + 6*a*b*d^2*x + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2)/((b*x + a)^6*b^3)