3.1367 \(\int \frac{(a+b x)^4}{(c+d x)^8} \, dx\)

Optimal. Leaf size=89 \[ \frac{b^2 (a+b x)^5}{105 (c+d x)^5 (b c-a d)^3}+\frac{b (a+b x)^5}{21 (c+d x)^6 (b c-a d)^2}+\frac{(a+b x)^5}{7 (c+d x)^7 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0192818, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{b^2 (a+b x)^5}{105 (c+d x)^5 (b c-a d)^3}+\frac{b (a+b x)^5}{21 (c+d x)^6 (b c-a d)^2}+\frac{(a+b x)^5}{7 (c+d x)^7 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(c + d*x)^8,x]

[Out]

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

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)^4}{(c+d x)^8} \, dx &=\frac{(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac{(2 b) \int \frac{(a+b x)^4}{(c+d x)^7} \, dx}{7 (b c-a d)}\\ &=\frac{(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac{b (a+b x)^5}{21 (b c-a d)^2 (c+d x)^6}+\frac{b^2 \int \frac{(a+b x)^4}{(c+d x)^6} \, dx}{21 (b c-a d)^2}\\ &=\frac{(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac{b (a+b x)^5}{21 (b c-a d)^2 (c+d x)^6}+\frac{b^2 (a+b x)^5}{105 (b c-a d)^3 (c+d x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0478935, size = 144, normalized size = 1.62 \[ -\frac{6 a^2 b^2 d^2 \left (c^2+7 c d x+21 d^2 x^2\right )+10 a^3 b d^3 (c+7 d x)+15 a^4 d^4+3 a b^3 d \left (7 c^2 d x+c^3+21 c d^2 x^2+35 d^3 x^3\right )+b^4 \left (21 c^2 d^2 x^2+7 c^3 d x+c^4+35 c d^3 x^3+35 d^4 x^4\right )}{105 d^5 (c+d x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x)^8,x]

[Out]

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

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Maple [B]  time = 0.005, size = 186, normalized size = 2.1 \begin{align*} -{\frac{6\,{b}^{2} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{5\,{d}^{5} \left ( dx+c \right ) ^{5}}}-{\frac{{a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,{c}^{3}a{b}^{3}d+{b}^{4}{c}^{4}}{7\,{d}^{5} \left ( dx+c \right ) ^{7}}}-{\frac{2\,b \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{3\,{d}^{5} \left ( dx+c \right ) ^{6}}}-{\frac{{b}^{4}}{3\,{d}^{5} \left ( dx+c \right ) ^{3}}}-{\frac{{b}^{3} \left ( ad-bc \right ) }{{d}^{5} \left ( dx+c \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(d*x+c)^8,x)

[Out]

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

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Maxima [B]  time = 1.00428, size = 333, normalized size = 3.74 \begin{align*} -\frac{35 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 35 \,{\left (b^{4} c d^{3} + 3 \, a b^{3} d^{4}\right )} x^{3} + 21 \,{\left (b^{4} c^{2} d^{2} + 3 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} + 7 \,{\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} + 10 \, a^{3} b d^{4}\right )} x}{105 \,{\left (d^{12} x^{7} + 7 \, c d^{11} x^{6} + 21 \, c^{2} d^{10} x^{5} + 35 \, c^{3} d^{9} x^{4} + 35 \, c^{4} d^{8} x^{3} + 21 \, c^{5} d^{7} x^{2} + 7 \, c^{6} d^{6} x + c^{7} d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/105*(35*b^4*d^4*x^4 + b^4*c^4 + 3*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 15*a^4*d^4 + 35*(b^4*c
*d^3 + 3*a*b^3*d^4)*x^3 + 21*(b^4*c^2*d^2 + 3*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*x^2 + 7*(b^4*c^3*d + 3*a*b^3*c^2*d^
2 + 6*a^2*b^2*c*d^3 + 10*a^3*b*d^4)*x)/(d^12*x^7 + 7*c*d^11*x^6 + 21*c^2*d^10*x^5 + 35*c^3*d^9*x^4 + 35*c^4*d^
8*x^3 + 21*c^5*d^7*x^2 + 7*c^6*d^6*x + c^7*d^5)

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Fricas [B]  time = 1.48325, size = 512, normalized size = 5.75 \begin{align*} -\frac{35 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 35 \,{\left (b^{4} c d^{3} + 3 \, a b^{3} d^{4}\right )} x^{3} + 21 \,{\left (b^{4} c^{2} d^{2} + 3 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} + 7 \,{\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} + 10 \, a^{3} b d^{4}\right )} x}{105 \,{\left (d^{12} x^{7} + 7 \, c d^{11} x^{6} + 21 \, c^{2} d^{10} x^{5} + 35 \, c^{3} d^{9} x^{4} + 35 \, c^{4} d^{8} x^{3} + 21 \, c^{5} d^{7} x^{2} + 7 \, c^{6} d^{6} x + c^{7} d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/105*(35*b^4*d^4*x^4 + b^4*c^4 + 3*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 15*a^4*d^4 + 35*(b^4*c
*d^3 + 3*a*b^3*d^4)*x^3 + 21*(b^4*c^2*d^2 + 3*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*x^2 + 7*(b^4*c^3*d + 3*a*b^3*c^2*d^
2 + 6*a^2*b^2*c*d^3 + 10*a^3*b*d^4)*x)/(d^12*x^7 + 7*c*d^11*x^6 + 21*c^2*d^10*x^5 + 35*c^3*d^9*x^4 + 35*c^4*d^
8*x^3 + 21*c^5*d^7*x^2 + 7*c^6*d^6*x + c^7*d^5)

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Sympy [B]  time = 11.6131, size = 264, normalized size = 2.97 \begin{align*} - \frac{15 a^{4} d^{4} + 10 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} + 3 a b^{3} c^{3} d + b^{4} c^{4} + 35 b^{4} d^{4} x^{4} + x^{3} \left (105 a b^{3} d^{4} + 35 b^{4} c d^{3}\right ) + x^{2} \left (126 a^{2} b^{2} d^{4} + 63 a b^{3} c d^{3} + 21 b^{4} c^{2} d^{2}\right ) + x \left (70 a^{3} b d^{4} + 42 a^{2} b^{2} c d^{3} + 21 a b^{3} c^{2} d^{2} + 7 b^{4} c^{3} d\right )}{105 c^{7} d^{5} + 735 c^{6} d^{6} x + 2205 c^{5} d^{7} x^{2} + 3675 c^{4} d^{8} x^{3} + 3675 c^{3} d^{9} x^{4} + 2205 c^{2} d^{10} x^{5} + 735 c d^{11} x^{6} + 105 d^{12} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(d*x+c)**8,x)

[Out]

-(15*a**4*d**4 + 10*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 + 3*a*b**3*c**3*d + b**4*c**4 + 35*b**4*d**4*x**4 +
x**3*(105*a*b**3*d**4 + 35*b**4*c*d**3) + x**2*(126*a**2*b**2*d**4 + 63*a*b**3*c*d**3 + 21*b**4*c**2*d**2) + x
*(70*a**3*b*d**4 + 42*a**2*b**2*c*d**3 + 21*a*b**3*c**2*d**2 + 7*b**4*c**3*d))/(105*c**7*d**5 + 735*c**6*d**6*
x + 2205*c**5*d**7*x**2 + 3675*c**4*d**8*x**3 + 3675*c**3*d**9*x**4 + 2205*c**2*d**10*x**5 + 735*c*d**11*x**6
+ 105*d**12*x**7)

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Giac [B]  time = 1.06834, size = 248, normalized size = 2.79 \begin{align*} -\frac{35 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c d^{3} x^{3} + 105 \, a b^{3} d^{4} x^{3} + 21 \, b^{4} c^{2} d^{2} x^{2} + 63 \, a b^{3} c d^{3} x^{2} + 126 \, a^{2} b^{2} d^{4} x^{2} + 7 \, b^{4} c^{3} d x + 21 \, a b^{3} c^{2} d^{2} x + 42 \, a^{2} b^{2} c d^{3} x + 70 \, a^{3} b d^{4} x + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4}}{105 \,{\left (d x + c\right )}^{7} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^8,x, algorithm="giac")

[Out]

-1/105*(35*b^4*d^4*x^4 + 35*b^4*c*d^3*x^3 + 105*a*b^3*d^4*x^3 + 21*b^4*c^2*d^2*x^2 + 63*a*b^3*c*d^3*x^2 + 126*
a^2*b^2*d^4*x^2 + 7*b^4*c^3*d*x + 21*a*b^3*c^2*d^2*x + 42*a^2*b^2*c*d^3*x + 70*a^3*b*d^4*x + b^4*c^4 + 3*a*b^3
*c^3*d + 6*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 15*a^4*d^4)/((d*x + c)^7*d^5)