3.1366 \(\int \frac{(a+b x)^5}{(c+d x)^8} \, dx\)

Optimal. Leaf size=58 \[ \frac{b (a+b x)^6}{42 (c+d x)^6 (b c-a d)^2}+\frac{(a+b x)^6}{7 (c+d x)^7 (b c-a d)} \]

[Out]

(a + b*x)^6/(7*(b*c - a*d)*(c + d*x)^7) + (b*(a + b*x)^6)/(42*(b*c - a*d)^2*(c + d*x)^6)

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Rubi [A]  time = 0.0112179, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{b (a+b x)^6}{42 (c+d x)^6 (b c-a d)^2}+\frac{(a+b x)^6}{7 (c+d x)^7 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(c + d*x)^8,x]

[Out]

(a + b*x)^6/(7*(b*c - a*d)*(c + d*x)^7) + (b*(a + b*x)^6)/(42*(b*c - a*d)^2*(c + d*x)^6)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{(c+d x)^8} \, dx &=\frac{(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac{b \int \frac{(a+b x)^5}{(c+d x)^7} \, dx}{7 (b c-a d)}\\ &=\frac{(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac{b (a+b x)^6}{42 (b c-a d)^2 (c+d x)^6}\\ \end{align*}

Mathematica [B]  time = 0.0577406, size = 205, normalized size = 3.53 \[ -\frac{3 a^2 b^3 d^2 \left (7 c^2 d x+c^3+21 c d^2 x^2+35 d^3 x^3\right )+4 a^3 b^2 d^3 \left (c^2+7 c d x+21 d^2 x^2\right )+5 a^4 b d^4 (c+7 d x)+6 a^5 d^5+2 a b^4 d \left (21 c^2 d^2 x^2+7 c^3 d x+c^4+35 c d^3 x^3+35 d^4 x^4\right )+b^5 \left (21 c^3 d^2 x^2+35 c^2 d^3 x^3+7 c^4 d x+c^5+35 c d^4 x^4+21 d^5 x^5\right )}{42 d^6 (c+d x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(c + d*x)^8,x]

[Out]

-(6*a^5*d^5 + 5*a^4*b*d^4*(c + 7*d*x) + 4*a^3*b^2*d^3*(c^2 + 7*c*d*x + 21*d^2*x^2) + 3*a^2*b^3*d^2*(c^3 + 7*c^
2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3) + 2*a*b^4*d*(c^4 + 7*c^3*d*x + 21*c^2*d^2*x^2 + 35*c*d^3*x^3 + 35*d^4*x^4)
+ b^5*(c^5 + 7*c^4*d*x + 21*c^3*d^2*x^2 + 35*c^2*d^3*x^3 + 35*c*d^4*x^4 + 21*d^5*x^5))/(42*d^6*(c + d*x)^7)

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Maple [B]  time = 0.005, size = 265, normalized size = 4.6 \begin{align*} -{\frac{{a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5}}{7\,{d}^{6} \left ( dx+c \right ) ^{7}}}-{\frac{{b}^{5}}{2\,{d}^{6} \left ( dx+c \right ) ^{2}}}-2\,{\frac{{b}^{2} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{{d}^{6} \left ( dx+c \right ) ^{5}}}-{\frac{5\,{b}^{3} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{2\,{d}^{6} \left ( dx+c \right ) ^{4}}}-{\frac{5\,{b}^{4} \left ( ad-bc \right ) }{3\,{d}^{6} \left ( dx+c \right ) ^{3}}}-{\frac{5\,b \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{6\,{d}^{6} \left ( dx+c \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(d*x+c)^8,x)

[Out]

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

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Maxima [B]  time = 1.01833, size = 440, normalized size = 7.59 \begin{align*} -\frac{21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \,{\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \,{\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \,{\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \,{\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \,{\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/42*(21*b^5*d^5*x^5 + b^5*c^5 + 2*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 + 4*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + 6*a^
5*d^5 + 35*(b^5*c*d^4 + 2*a*b^4*d^5)*x^4 + 35*(b^5*c^2*d^3 + 2*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 21*(b^5*c^3*
d^2 + 2*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 + 4*a^3*b^2*d^5)*x^2 + 7*(b^5*c^4*d + 2*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*
d^3 + 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x)/(d^13*x^7 + 7*c*d^12*x^6 + 21*c^2*d^11*x^5 + 35*c^3*d^10*x^4 + 35*c^4*
d^9*x^3 + 21*c^5*d^8*x^2 + 7*c^6*d^7*x + c^7*d^6)

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Fricas [B]  time = 1.74847, size = 663, normalized size = 11.43 \begin{align*} -\frac{21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \,{\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \,{\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \,{\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \,{\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \,{\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*d^5*x^5 + b^5*c^5 + 2*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 + 4*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + 6*a^
5*d^5 + 35*(b^5*c*d^4 + 2*a*b^4*d^5)*x^4 + 35*(b^5*c^2*d^3 + 2*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 21*(b^5*c^3*
d^2 + 2*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 + 4*a^3*b^2*d^5)*x^2 + 7*(b^5*c^4*d + 2*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*
d^3 + 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x)/(d^13*x^7 + 7*c*d^12*x^6 + 21*c^2*d^11*x^5 + 35*c^3*d^10*x^4 + 35*c^4*
d^9*x^3 + 21*c^5*d^8*x^2 + 7*c^6*d^7*x + c^7*d^6)

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Sympy [B]  time = 52.1013, size = 348, normalized size = 6. \begin{align*} - \frac{6 a^{5} d^{5} + 5 a^{4} b c d^{4} + 4 a^{3} b^{2} c^{2} d^{3} + 3 a^{2} b^{3} c^{3} d^{2} + 2 a b^{4} c^{4} d + b^{5} c^{5} + 21 b^{5} d^{5} x^{5} + x^{4} \left (70 a b^{4} d^{5} + 35 b^{5} c d^{4}\right ) + x^{3} \left (105 a^{2} b^{3} d^{5} + 70 a b^{4} c d^{4} + 35 b^{5} c^{2} d^{3}\right ) + x^{2} \left (84 a^{3} b^{2} d^{5} + 63 a^{2} b^{3} c d^{4} + 42 a b^{4} c^{2} d^{3} + 21 b^{5} c^{3} d^{2}\right ) + x \left (35 a^{4} b d^{5} + 28 a^{3} b^{2} c d^{4} + 21 a^{2} b^{3} c^{2} d^{3} + 14 a b^{4} c^{3} d^{2} + 7 b^{5} c^{4} d\right )}{42 c^{7} d^{6} + 294 c^{6} d^{7} x + 882 c^{5} d^{8} x^{2} + 1470 c^{4} d^{9} x^{3} + 1470 c^{3} d^{10} x^{4} + 882 c^{2} d^{11} x^{5} + 294 c d^{12} x^{6} + 42 d^{13} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(d*x+c)**8,x)

[Out]

-(6*a**5*d**5 + 5*a**4*b*c*d**4 + 4*a**3*b**2*c**2*d**3 + 3*a**2*b**3*c**3*d**2 + 2*a*b**4*c**4*d + b**5*c**5
+ 21*b**5*d**5*x**5 + x**4*(70*a*b**4*d**5 + 35*b**5*c*d**4) + x**3*(105*a**2*b**3*d**5 + 70*a*b**4*c*d**4 + 3
5*b**5*c**2*d**3) + x**2*(84*a**3*b**2*d**5 + 63*a**2*b**3*c*d**4 + 42*a*b**4*c**2*d**3 + 21*b**5*c**3*d**2) +
x*(35*a**4*b*d**5 + 28*a**3*b**2*c*d**4 + 21*a**2*b**3*c**2*d**3 + 14*a*b**4*c**3*d**2 + 7*b**5*c**4*d))/(42*
c**7*d**6 + 294*c**6*d**7*x + 882*c**5*d**8*x**2 + 1470*c**4*d**9*x**3 + 1470*c**3*d**10*x**4 + 882*c**2*d**11
*x**5 + 294*c*d**12*x**6 + 42*d**13*x**7)

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Giac [B]  time = 1.06566, size = 366, normalized size = 6.31 \begin{align*} -\frac{21 \, b^{5} d^{5} x^{5} + 35 \, b^{5} c d^{4} x^{4} + 70 \, a b^{4} d^{5} x^{4} + 35 \, b^{5} c^{2} d^{3} x^{3} + 70 \, a b^{4} c d^{4} x^{3} + 105 \, a^{2} b^{3} d^{5} x^{3} + 21 \, b^{5} c^{3} d^{2} x^{2} + 42 \, a b^{4} c^{2} d^{3} x^{2} + 63 \, a^{2} b^{3} c d^{4} x^{2} + 84 \, a^{3} b^{2} d^{5} x^{2} + 7 \, b^{5} c^{4} d x + 14 \, a b^{4} c^{3} d^{2} x + 21 \, a^{2} b^{3} c^{2} d^{3} x + 28 \, a^{3} b^{2} c d^{4} x + 35 \, a^{4} b d^{5} x + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5}}{42 \,{\left (d x + c\right )}^{7} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^8,x, algorithm="giac")

[Out]

-1/42*(21*b^5*d^5*x^5 + 35*b^5*c*d^4*x^4 + 70*a*b^4*d^5*x^4 + 35*b^5*c^2*d^3*x^3 + 70*a*b^4*c*d^4*x^3 + 105*a^
2*b^3*d^5*x^3 + 21*b^5*c^3*d^2*x^2 + 42*a*b^4*c^2*d^3*x^2 + 63*a^2*b^3*c*d^4*x^2 + 84*a^3*b^2*d^5*x^2 + 7*b^5*
c^4*d*x + 14*a*b^4*c^3*d^2*x + 21*a^2*b^3*c^2*d^3*x + 28*a^3*b^2*c*d^4*x + 35*a^4*b*d^5*x + b^5*c^5 + 2*a*b^4*
c^4*d + 3*a^2*b^3*c^3*d^2 + 4*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + 6*a^5*d^5)/((d*x + c)^7*d^6)