3.14.66 \(\int \frac {(a+b x)^5}{(c+d x)^8} \, dx\) [1366]

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

[Out]

1/7*(b*x+a)^6/(-a*d+b*c)/(d*x+c)^7+1/42*b*(b*x+a)^6/(-a*d+b*c)^2/(d*x+c)^6

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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 = {47, 37} \begin {gather*} \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)} \end {gather*}

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 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 47

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)^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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(58)=116\).
time = 0.04, size = 205, normalized size = 3.53 \begin {gather*} -\frac {6 a^5 d^5+5 a^4 b d^4 (c+7 d x)+4 a^3 b^2 d^3 \left (c^2+7 c d x+21 d^2 x^2\right )+3 a^2 b^3 d^2 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+2 a b^4 d \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 )+b^5 \left (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\right )}{42 d^6 (c+d x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(c + d*x)^8,x]

[Out]

-1/42*(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))/(d^6*(c + d*x)^7)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(310\) vs. \(2(58)=116\).
time = 39.22, size = 307, normalized size = 5.29 \begin {gather*} -\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+7 b d x \left (5 a^4 d^4+4 a^3 b c d^3+3 a^2 b^2 c^2 d^2+2 a b^3 c^3 d+b^4 c^4\right )+21 b^2 d^2 x^2 \left (4 a^3 d^3+3 a^2 b c d^2+2 a b^2 c^2 d+b^3 c^3\right )+35 b^3 d^3 x^3 \left (3 a^2 d^2+2 a b c d+b^2 c^2\right )+35 b^4 d^4 x^4 \left (2 a d+b c\right )+21 b^5 d^5 x^5}{42 d^6 \left (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\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(264\) vs. \(2(54)=108\).
time = 0.16, size = 265, normalized size = 4.57

method	result	size
risch	\(\frac {-\frac {b^{5} x^{5}}{2 d}-\frac {5 b^{4} \left (2 a d +b c \right ) x^{4}}{6 d^{2}}-\frac {5 b^{3} \left (3 a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) x^{3}}{6 d^{3}}-\frac {b^{2} \left (4 a^{3} d^{3}+3 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{2}}{2 d^{4}}-\frac {b \left (5 a^{4} d^{4}+4 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{6 d^{5}}-\frac {6 a^{5} d^{5}+5 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+2 a \,b^{4} c^{4} d +b^{5} c^{5}}{42 d^{6}}}{\left (d x +c \right )^{7}}\)	\(246\)
default	\(-\frac {5 b^{4} \left (a d -b c \right )}{3 d^{6} \left (d x +c \right )^{3}}-\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{6 d^{6} \left (d x +c \right )^{6}}-\frac {2 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{6} \left (d x +c \right )^{5}}-\frac {5 b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{6} \left (d x +c \right )^{4}}-\frac {b^{5}}{2 d^{6} \left (d x +c \right )^{2}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{7 d^{6} \left (d x +c \right )^{7}}\)	\(265\)
norman	\(\frac {-\frac {b^{5} x^{5}}{2 d}-\frac {5 \left (2 a \,b^{4} d^{2}+b^{5} c d \right ) x^{4}}{6 d^{3}}-\frac {5 \left (3 a^{2} b^{3} d^{3}+2 a \,b^{4} c \,d^{2}+b^{5} c^{2} d \right ) x^{3}}{6 d^{4}}-\frac {\left (4 a^{3} b^{2} d^{4}+3 a^{2} b^{3} c \,d^{3}+2 a \,b^{4} c^{2} d^{2}+b^{5} c^{3} d \right ) x^{2}}{2 d^{5}}-\frac {\left (5 a^{4} b \,d^{5}+4 a^{3} b^{2} c \,d^{4}+3 a^{2} b^{3} c^{2} d^{3}+2 a \,b^{4} c^{3} d^{2}+b^{5} c^{4} d \right ) x}{6 d^{6}}-\frac {6 a^{5} d^{6}+5 a^{4} b c \,d^{5}+4 a^{3} b^{2} c^{2} d^{4}+3 a^{2} b^{3} c^{3} d^{3}+2 a \,b^{4} c^{4} d^{2}+b^{5} c^{5} d}{42 d^{7}}}{\left (d x +c \right )^{7}}\)	\(269\)
gosper	\(-\frac {21 b^{5} x^{5} d^{5}+70 a \,b^{4} d^{5} x^{4}+35 b^{5} c \,d^{4} x^{4}+105 a^{2} b^{3} d^{5} x^{3}+70 a \,b^{4} c \,d^{4} x^{3}+35 b^{5} c^{2} d^{3} x^{3}+84 a^{3} b^{2} d^{5} x^{2}+63 a^{2} b^{3} c \,d^{4} x^{2}+42 a \,b^{4} c^{2} d^{3} x^{2}+21 b^{5} c^{3} d^{2} x^{2}+35 a^{4} b \,d^{5} x +28 a^{3} b^{2} c \,d^{4} x +21 a^{2} b^{3} c^{2} d^{3} x +14 a \,b^{4} c^{3} d^{2} x +7 b^{5} c^{4} d x +6 a^{5} d^{5}+5 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+2 a \,b^{4} c^{4} d +b^{5} c^{5}}{42 d^{6} \left (d x +c \right )^{7}}\)	\(272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (54) = 108\).
time = 0.29, size = 326, normalized size = 5.62 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (54) = 108\).
time = 0.29, size = 326, normalized size = 5.62 \begin {gather*} -\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 {gather*}

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 [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (54) = 108\).
time = 0.00, size = 294, normalized size = 5.07 \begin {gather*} \frac {-21 x^{5} b^{5} d^{5}-35 x^{4} b^{5} d^{4} c-70 x^{4} b^{4} a d^{5}-35 x^{3} b^{5} d^{3} c^{2}-70 x^{3} b^{4} a d^{4} c-105 x^{3} b^{3} a^{2} d^{5}-21 x^{2} b^{5} d^{2} c^{3}-42 x^{2} b^{4} a d^{3} c^{2}-63 x^{2} b^{3} a^{2} d^{4} c-84 x^{2} b^{2} a^{3} d^{5}-7 x b^{5} d c^{4}-14 x b^{4} a d^{2} c^{3}-21 x b^{3} a^{2} d^{3} c^{2}-28 x b^{2} a^{3} d^{4} c-35 x b a^{4} d^{5}-b^{5} c^{5}-2 b^{4} a d c^{4}-3 b^{3} a^{2} d^{2} c^{3}-4 b^{2} a^{3} d^{3} c^{2}-5 b a^{4} d^{4} c-6 a^{5} d^{5}}{42 d^{6} \left (x d+c\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Mupad [B]
time = 0.28, size = 39, normalized size = 0.67 \begin {gather*} \frac {{\left (a+b\,x\right )}^6\,\left (7\,b\,c-6\,a\,d+b\,d\,x\right )}{42\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(c + d*x)^8,x)

[Out]

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

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