3.1.39 \(\int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx\) [39]

Optimal. Leaf size=41 \[ -\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6} \]

[Out]

-1/7*c^5*(-b*x+a)^6/x^7-4/21*b*c^5*(-b*x+a)^6/a/x^6

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \begin {gather*} -\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

-1/7*(c^5*(a - b*x)^6)/x^7 - (4*b*c^5*(a - b*x)^6)/(21*a*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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx &=-\frac {c^5 (a-b x)^6}{7 x^7}+\frac {1}{7} (8 b) \int \frac {(a c-b c x)^5}{x^7} \, dx\\ &=-\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 66, normalized size = 1.61 \begin {gather*} c^5 \left (-\frac {a^6}{7 x^7}+\frac {2 a^5 b}{3 x^6}-\frac {a^4 b^2}{x^5}+\frac {5 a^2 b^4}{3 x^3}-\frac {2 a b^5}{x^2}+\frac {b^6}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 61, normalized size = 1.49

method	result	size
gosper	\(-\frac {c^{5} \left (-21 b^{6} x^{6}+42 a \,b^{5} x^{5}-35 a^{2} b^{4} x^{4}+21 a^{4} b^{2} x^{2}-14 a^{5} b x +3 a^{6}\right )}{21 x^{7}}\)	\(61\)
default	\(c^{5} \left (\frac {b^{6}}{x}-\frac {a^{4} b^{2}}{x^{5}}+\frac {2 a^{5} b}{3 x^{6}}-\frac {2 a \,b^{5}}{x^{2}}+\frac {5 a^{2} b^{4}}{3 x^{3}}-\frac {a^{6}}{7 x^{7}}\right )\)	\(61\)
norman	\(\frac {b^{6} c^{5} x^{6}-\frac {1}{7} a^{6} c^{5}-2 a \,b^{5} c^{5} x^{5}+\frac {5}{3} a^{2} b^{4} c^{5} x^{4}-a^{4} b^{2} c^{5} x^{2}+\frac {2}{3} a^{5} b \,c^{5} x}{x^{7}}\)	\(74\)
risch	\(\frac {b^{6} c^{5} x^{6}-\frac {1}{7} a^{6} c^{5}-2 a \,b^{5} c^{5} x^{5}+\frac {5}{3} a^{2} b^{4} c^{5} x^{4}-a^{4} b^{2} c^{5} x^{2}+\frac {2}{3} a^{5} b \,c^{5} x}{x^{7}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 75, normalized size = 1.83 \begin {gather*} \frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

________________________________________________________________________________________

Fricas [A]
time = 1.01, size = 75, normalized size = 1.83 \begin {gather*} \frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (36) = 72\).
time = 0.21, size = 82, normalized size = 2.00 \begin {gather*} - \frac {3 a^{6} c^{5} - 14 a^{5} b c^{5} x + 21 a^{4} b^{2} c^{5} x^{2} - 35 a^{2} b^{4} c^{5} x^{4} + 42 a b^{5} c^{5} x^{5} - 21 b^{6} c^{5} x^{6}}{21 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a**6*c**5 - 14*a**5*b*c**5*x + 21*a**4*b**2*c**5*x**2 - 35*a**2*b**4*c**5*x**4 + 42*a*b**5*c**5*x**5 - 21*
b**6*c**5*x**6)/(21*x**7)

________________________________________________________________________________________

Giac [A]
time = 1.76, size = 75, normalized size = 1.83 \begin {gather*} \frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

________________________________________________________________________________________

Mupad [B]
time = 0.29, size = 74, normalized size = 1.80 \begin {gather*} -\frac {\frac {a^6\,c^5}{7}-\frac {2\,a^5\,b\,c^5\,x}{3}+a^4\,b^2\,c^5\,x^2-\frac {5\,a^2\,b^4\,c^5\,x^4}{3}+2\,a\,b^5\,c^5\,x^5-b^6\,c^5\,x^6}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________