3.14.68 \(\int \frac {(a+b x)^3}{(c+d x)^8} \, dx\) [1368]

Optimal. Leaf size=92 \[ \frac {(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac {b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac {3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac {b^3}{4 d^4 (c+d x)^4} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac {b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac {(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac {b^3}{4 d^4 (c+d x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(c + d*x)^8,x]

[Out]

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

Rule 45

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

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Mathematica [A]
time = 0.02, size = 94, normalized size = 1.02 \begin {gather*} -\frac {20 a^3 d^3+10 a^2 b d^2 (c+7 d x)+4 a b^2 d \left (c^2+7 c d x+21 d^2 x^2\right )+b^3 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )}{140 d^4 (c+d x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/(c + d*x)^8,x]

[Out]

-1/140*(20*a^3*d^3 + 10*a^2*b*d^2*(c + 7*d*x) + 4*a*b^2*d*(c^2 + 7*c*d*x + 21*d^2*x^2) + b^3*(c^3 + 7*c^2*d*x
+ 21*c*d^2*x^2 + 35*d^3*x^3))/(d^4*(c + d*x)^7)

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Maple [A]
time = 0.14, size = 122, normalized size = 1.33

method	result	size
risch	\(\frac {-\frac {b^{3} x^{3}}{4 d}-\frac {3 b^{2} \left (4 a d +b c \right ) x^{2}}{20 d^{2}}-\frac {b \left (10 a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x}{20 d^{3}}-\frac {20 a^{3} d^{3}+10 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d +b^{3} c^{3}}{140 d^{4}}}{\left (d x +c \right )^{7}}\)	\(110\)
gosper	\(-\frac {35 b^{3} x^{3} d^{3}+84 a \,b^{2} d^{3} x^{2}+21 b^{3} c \,d^{2} x^{2}+70 a^{2} b \,d^{3} x +28 a \,b^{2} c \,d^{2} x +7 b^{3} c^{2} d x +20 a^{3} d^{3}+10 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d +b^{3} c^{3}}{140 d^{4} \left (d x +c \right )^{7}}\)	\(115\)
default	\(-\frac {b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{4} \left (d x +c \right )^{6}}-\frac {3 b^{2} \left (a d -b c \right )}{5 d^{4} \left (d x +c \right )^{5}}-\frac {b^{3}}{4 d^{4} \left (d x +c \right )^{4}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{7 d^{4} \left (d x +c \right )^{7}}\)	\(122\)
norman	\(\frac {-\frac {b^{3} x^{3}}{4 d}-\frac {3 \left (4 a \,b^{2} d^{4}+b^{3} c \,d^{3}\right ) x^{2}}{20 d^{5}}-\frac {\left (10 a^{2} b \,d^{5}+4 a \,b^{2} c \,d^{4}+b^{3} c^{2} d^{3}\right ) x}{20 d^{6}}-\frac {20 a^{3} d^{6}+10 a^{2} b c \,d^{5}+4 a \,b^{2} c^{2} d^{4}+b^{3} c^{3} d^{3}}{140 d^{7}}}{\left (d x +c \right )^{7}}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)^8,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (84) = 168\).
time = 0.28, size = 182, normalized size = 1.98 \begin {gather*} -\frac {35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \, {\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \, {\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*d^3*x^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3 + 21*(b^3*c*d^2 + 4*a*b^2*d^3)*
x^2 + 7*(b^3*c^2*d + 4*a*b^2*c*d^2 + 10*a^2*b*d^3)*x)/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 + 35*c^3*d^8*x
^4 + 35*c^4*d^7*x^3 + 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (84) = 168\).
time = 0.80, size = 182, normalized size = 1.98 \begin {gather*} -\frac {35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \, {\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \, {\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*d^3*x^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3 + 21*(b^3*c*d^2 + 4*a*b^2*d^3)*
x^2 + 7*(b^3*c^2*d + 4*a*b^2*c*d^2 + 10*a^2*b*d^3)*x)/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 + 35*c^3*d^8*x
^4 + 35*c^4*d^7*x^3 + 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (80) = 160\).
time = 1.94, size = 196, normalized size = 2.13 \begin {gather*} \frac {- 20 a^{3} d^{3} - 10 a^{2} b c d^{2} - 4 a b^{2} c^{2} d - b^{3} c^{3} - 35 b^{3} d^{3} x^{3} + x^{2} \left (- 84 a b^{2} d^{3} - 21 b^{3} c d^{2}\right ) + x \left (- 70 a^{2} b d^{3} - 28 a b^{2} c d^{2} - 7 b^{3} c^{2} d\right )}{140 c^{7} d^{4} + 980 c^{6} d^{5} x + 2940 c^{5} d^{6} x^{2} + 4900 c^{4} d^{7} x^{3} + 4900 c^{3} d^{8} x^{4} + 2940 c^{2} d^{9} x^{5} + 980 c d^{10} x^{6} + 140 d^{11} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/(d*x+c)**8,x)

[Out]

(-20*a**3*d**3 - 10*a**2*b*c*d**2 - 4*a*b**2*c**2*d - b**3*c**3 - 35*b**3*d**3*x**3 + x**2*(-84*a*b**2*d**3 -
21*b**3*c*d**2) + x*(-70*a**2*b*d**3 - 28*a*b**2*c*d**2 - 7*b**3*c**2*d))/(140*c**7*d**4 + 980*c**6*d**5*x + 2
940*c**5*d**6*x**2 + 4900*c**4*d**7*x**3 + 4900*c**3*d**8*x**4 + 2940*c**2*d**9*x**5 + 980*c*d**10*x**6 + 140*
d**11*x**7)

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Giac [A]
time = 0.85, size = 114, normalized size = 1.24 \begin {gather*} -\frac {35 \, b^{3} d^{3} x^{3} + 21 \, b^{3} c d^{2} x^{2} + 84 \, a b^{2} d^{3} x^{2} + 7 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 70 \, a^{2} b d^{3} x + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3}}{140 \, {\left (d x + c\right )}^{7} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*d^3*x^3 + 21*b^3*c*d^2*x^2 + 84*a*b^2*d^3*x^2 + 7*b^3*c^2*d*x + 28*a*b^2*c*d^2*x + 70*a^2*b*d^3
*x + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3)/((d*x + c)^7*d^4)

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Mupad [B]
time = 0.10, size = 176, normalized size = 1.91 \begin {gather*} -\frac {\frac {20\,a^3\,d^3+10\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d+b^3\,c^3}{140\,d^4}+\frac {b^3\,x^3}{4\,d}+\frac {b\,x\,\left (10\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{20\,d^3}+\frac {3\,b^2\,x^2\,\left (4\,a\,d+b\,c\right )}{20\,d^2}}{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)^3/(c + d*x)^8,x)

[Out]

-((20*a^3*d^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2)/(140*d^4) + (b^3*x^3)/(4*d) + (b*x*(10*a^2*d^2 + b^2
*c^2 + 4*a*b*c*d))/(20*d^3) + (3*b^2*x^2*(4*a*d + b*c))/(20*d^2))/(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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