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

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

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Rubi [A]  time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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]

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])

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*}

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Mathematica [A]  time = 0.05, size = 144, normalized size = 1.62 \begin {gather*} -\frac {15 a^4 d^4+10 a^3 b d^3 (c+7 d x)+6 a^2 b^2 d^2 \left (c^2+7 c d x+21 d^2 x^2\right )+3 a b^3 d \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+b^4 \left (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\right )}{105 d^5 (c+d x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x)^8,x]

[Out]

-1/105*(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))/(
d^5*(c + d*x)^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^4}{(c+d x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^4/(c + d*x)^8,x]

[Out]

IntegrateAlgebraic[(a + b*x)^4/(c + d*x)^8, x]

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fricas [B]  time = 1.06, size = 247, normalized size = 2.78 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 1.29, size = 184, normalized size = 2.07 \begin {gather*} -\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 {gather*}

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)

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maple [B]  time = 0.01, size = 186, normalized size = 2.09 \begin {gather*} -\frac {b^{4}}{3 \left (d x +c \right )^{3} d^{5}}-\frac {\left (a d -b c \right ) b^{3}}{\left (d x +c \right )^{4} d^{5}}-\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2}}{5 \left (d x +c \right )^{5} d^{5}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{3 \left (d x +c \right )^{6} d^{5}}-\frac {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}}{7 \left (d x +c \right )^{7} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.52, size = 247, normalized size = 2.78 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.11, size = 237, normalized size = 2.66 \begin {gather*} -\frac {\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}{105\,d^5}+\frac {b^4\,x^4}{3\,d}+\frac {b^3\,x^3\,\left (3\,a\,d+b\,c\right )}{3\,d^2}+\frac {b\,x\,\left (10\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{15\,d^4}+\frac {b^2\,x^2\,\left (6\,a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{5\,d^3}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(c + d*x)^8,x)

[Out]

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

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sympy [B]  time = 9.64, size = 267, normalized size = 3.00 \begin {gather*} \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 {gather*}

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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