3.1048 \(\int \frac{(a+b x)^2}{(a c-b c x)^7} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 a^2}{3 b c^7 (a-b x)^6}-\frac{4 a}{5 b c^7 (a-b x)^5}+\frac{1}{4 b c^7 (a-b x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0290148, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ \frac{2 a^2}{3 b c^7 (a-b x)^6}-\frac{4 a}{5 b c^7 (a-b x)^5}+\frac{1}{4 b c^7 (a-b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^7,x]

[Out]

(2*a^2)/(3*b*c^7*(a - b*x)^6) - (4*a)/(5*b*c^7*(a - b*x)^5) + 1/(4*b*c^7*(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{(a+b x)^2}{(a c-b c x)^7} \, dx &=\int \left (\frac{4 a^2}{c^7 (a-b x)^7}-\frac{4 a}{c^7 (a-b x)^6}+\frac{1}{c^7 (a-b x)^5}\right ) \, dx\\ &=\frac{2 a^2}{3 b c^7 (a-b x)^6}-\frac{4 a}{5 b c^7 (a-b x)^5}+\frac{1}{4 b c^7 (a-b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0148413, size = 37, normalized size = 0.63 \[ \frac{7 a^2+18 a b x+15 b^2 x^2}{60 b c^7 (a-b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a*c - b*c*x)^7,x]

[Out]

(7*a^2 + 18*a*b*x + 15*b^2*x^2)/(60*b*c^7*(a - b*x)^6)

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Maple [A]  time = 0.004, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{7}} \left ({\frac{1}{4\,b \left ( bx-a \right ) ^{4}}}+{\frac{4\,a}{5\,b \left ( bx-a \right ) ^{5}}}+{\frac{2\,{a}^{2}}{3\,b \left ( bx-a \right ) ^{6}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/(-b*c*x+a*c)^7,x)

[Out]

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

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Maxima [A]  time = 1.05498, size = 146, normalized size = 2.47 \begin{align*} \frac{15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \,{\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(15*b^2*x^2 + 18*a*b*x + 7*a^2)/(b^7*c^7*x^6 - 6*a*b^6*c^7*x^5 + 15*a^2*b^5*c^7*x^4 - 20*a^3*b^4*c^7*x^3
+ 15*a^4*b^3*c^7*x^2 - 6*a^5*b^2*c^7*x + a^6*b*c^7)

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Fricas [A]  time = 1.45674, size = 220, normalized size = 3.73 \begin{align*} \frac{15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \,{\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(15*b^2*x^2 + 18*a*b*x + 7*a^2)/(b^7*c^7*x^6 - 6*a*b^6*c^7*x^5 + 15*a^2*b^5*c^7*x^4 - 20*a^3*b^4*c^7*x^3
+ 15*a^4*b^3*c^7*x^2 - 6*a^5*b^2*c^7*x + a^6*b*c^7)

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Sympy [B]  time = 0.759735, size = 114, normalized size = 1.93 \begin{align*} \frac{7 a^{2} + 18 a b x + 15 b^{2} x^{2}}{60 a^{6} b c^{7} - 360 a^{5} b^{2} c^{7} x + 900 a^{4} b^{3} c^{7} x^{2} - 1200 a^{3} b^{4} c^{7} x^{3} + 900 a^{2} b^{5} c^{7} x^{4} - 360 a b^{6} c^{7} x^{5} + 60 b^{7} c^{7} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*a**2 + 18*a*b*x + 15*b**2*x**2)/(60*a**6*b*c**7 - 360*a**5*b**2*c**7*x + 900*a**4*b**3*c**7*x**2 - 1200*a**
3*b**4*c**7*x**3 + 900*a**2*b**5*c**7*x**4 - 360*a*b**6*c**7*x**5 + 60*b**7*c**7*x**6)

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Giac [A]  time = 1.06035, size = 49, normalized size = 0.83 \begin{align*} \frac{15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \,{\left (b x - a\right )}^{6} b c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(15*b^2*x^2 + 18*a*b*x + 7*a^2)/((b*x - a)^6*b*c^7)