3.1782 \(\int \frac{(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{d (b c-a d)}{3 b^3 (a+b x)^6}-\frac{(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac{d^2}{5 b^3 (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0413255, antiderivative size = 65, normalized size of antiderivative = 1., 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} \[ -\frac{d (b c-a d)}{3 b^3 (a+b x)^6}-\frac{(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac{d^2}{5 b^3 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^10,x]

[Out]

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

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]

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{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx &=\int \frac{(c+d x)^2}{(a+b x)^8} \, dx\\ &=\int \left (\frac{(b c-a d)^2}{b^2 (a+b x)^8}+\frac{2 d (b c-a d)}{b^2 (a+b x)^7}+\frac{d^2}{b^2 (a+b x)^6}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac{d (b c-a d)}{3 b^3 (a+b x)^6}-\frac{d^2}{5 b^3 (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0234481, size = 57, normalized size = 0.88 \[ -\frac{a^2 d^2+a b d (5 c+7 d x)+b^2 \left (15 c^2+35 c d x+21 d^2 x^2\right )}{105 b^3 (a+b x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^10,x]

[Out]

-(a^2*d^2 + a*b*d*(5*c + 7*d*x) + b^2*(15*c^2 + 35*c*d*x + 21*d^2*x^2))/(105*b^3*(a + b*x)^7)

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Maple [A]  time = 0.049, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{d}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{7\,{b}^{3} \left ( bx+a \right ) ^{7}}}+{\frac{ \left ( ad-bc \right ) d}{3\,{b}^{3} \left ( bx+a \right ) ^{6}}} \end{align*}

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)^10,x)

[Out]

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

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Maxima [B]  time = 1.05412, size = 177, normalized size = 2.72 \begin{align*} -\frac{21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \,{\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \,{\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \end{align*}

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)^10,x, algorithm="maxima")

[Out]

-1/105*(21*b^2*d^2*x^2 + 15*b^2*c^2 + 5*a*b*c*d + a^2*d^2 + 7*(5*b^2*c*d + a*b*d^2)*x)/(b^10*x^7 + 7*a*b^9*x^6
+ 21*a^2*b^8*x^5 + 35*a^3*b^7*x^4 + 35*a^4*b^6*x^3 + 21*a^5*b^5*x^2 + 7*a^6*b^4*x + a^7*b^3)

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Fricas [B]  time = 1.49137, size = 277, normalized size = 4.26 \begin{align*} -\frac{21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \,{\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \,{\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \end{align*}

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)^10,x, algorithm="fricas")

[Out]

-1/105*(21*b^2*d^2*x^2 + 15*b^2*c^2 + 5*a*b*c*d + a^2*d^2 + 7*(5*b^2*c*d + a*b*d^2)*x)/(b^10*x^7 + 7*a*b^9*x^6
+ 21*a^2*b^8*x^5 + 35*a^3*b^7*x^4 + 35*a^4*b^6*x^3 + 21*a^5*b^5*x^2 + 7*a^6*b^4*x + a^7*b^3)

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Sympy [B]  time = 2.81169, size = 139, normalized size = 2.14 \begin{align*} - \frac{a^{2} d^{2} + 5 a b c d + 15 b^{2} c^{2} + 21 b^{2} d^{2} x^{2} + x \left (7 a b d^{2} + 35 b^{2} c d\right )}{105 a^{7} b^{3} + 735 a^{6} b^{4} x + 2205 a^{5} b^{5} x^{2} + 3675 a^{4} b^{6} x^{3} + 3675 a^{3} b^{7} x^{4} + 2205 a^{2} b^{8} x^{5} + 735 a b^{9} x^{6} + 105 b^{10} x^{7}} \end{align*}

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)**10,x)

[Out]

-(a**2*d**2 + 5*a*b*c*d + 15*b**2*c**2 + 21*b**2*d**2*x**2 + x*(7*a*b*d**2 + 35*b**2*c*d))/(105*a**7*b**3 + 73
5*a**6*b**4*x + 2205*a**5*b**5*x**2 + 3675*a**4*b**6*x**3 + 3675*a**3*b**7*x**4 + 2205*a**2*b**8*x**5 + 735*a*
b**9*x**6 + 105*b**10*x**7)

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Giac [A]  time = 1.20101, size = 82, normalized size = 1.26 \begin{align*} -\frac{21 \, b^{2} d^{2} x^{2} + 35 \, b^{2} c d x + 7 \, a b d^{2} x + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2}}{105 \,{\left (b x + a\right )}^{7} b^{3}} \end{align*}

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)^10,x, algorithm="giac")

[Out]

-1/105*(21*b^2*d^2*x^2 + 35*b^2*c*d*x + 7*a*b*d^2*x + 15*b^2*c^2 + 5*a*b*c*d + a^2*d^2)/((b*x + a)^7*b^3)