3.2.81 \(\int \frac {x^6}{(a+b x)^3} \, dx\) [181]

Optimal. Leaf size=86 \[ -\frac {10 a^3 x}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {a x^3}{b^4}+\frac {x^4}{4 b^3}-\frac {a^6}{2 b^7 (a+b x)^2}+\frac {6 a^5}{b^7 (a+b x)}+\frac {15 a^4 \log (a+b x)}{b^7} \]

[Out]

-10*a^3*x/b^6+3*a^2*x^2/b^5-a*x^3/b^4+1/4*x^4/b^3-1/2*a^6/b^7/(b*x+a)^2+6*a^5/b^7/(b*x+a)+15*a^4*ln(b*x+a)/b^7

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Rubi [A]
time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} -\frac {a^6}{2 b^7 (a+b x)^2}+\frac {6 a^5}{b^7 (a+b x)}+\frac {15 a^4 \log (a+b x)}{b^7}-\frac {10 a^3 x}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {a x^3}{b^4}+\frac {x^4}{4 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x)^3,x]

[Out]

(-10*a^3*x)/b^6 + (3*a^2*x^2)/b^5 - (a*x^3)/b^4 + x^4/(4*b^3) - a^6/(2*b^7*(a + b*x)^2) + (6*a^5)/(b^7*(a + b*
x)) + (15*a^4*Log[a + b*x])/b^7

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 {x^6}{(a+b x)^3} \, dx &=\int \left (-\frac {10 a^3}{b^6}+\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{b^4}+\frac {x^3}{b^3}+\frac {a^6}{b^6 (a+b x)^3}-\frac {6 a^5}{b^6 (a+b x)^2}+\frac {15 a^4}{b^6 (a+b x)}\right ) \, dx\\ &=-\frac {10 a^3 x}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {a x^3}{b^4}+\frac {x^4}{4 b^3}-\frac {a^6}{2 b^7 (a+b x)^2}+\frac {6 a^5}{b^7 (a+b x)}+\frac {15 a^4 \log (a+b x)}{b^7}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 77, normalized size = 0.90 \begin {gather*} \frac {-40 a^3 b x+12 a^2 b^2 x^2-4 a b^3 x^3+b^4 x^4-\frac {2 a^6}{(a+b x)^2}+\frac {24 a^5}{a+b x}+60 a^4 \log (a+b x)}{4 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x)^3,x]

[Out]

(-40*a^3*b*x + 12*a^2*b^2*x^2 - 4*a*b^3*x^3 + b^4*x^4 - (2*a^6)/(a + b*x)^2 + (24*a^5)/(a + b*x) + 60*a^4*Log[
a + b*x])/(4*b^7)

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Mathics [A]
time = 2.39, size = 126, normalized size = 1.47 \begin {gather*} \frac {22 a^6+60 a^6 \text {Log}\left [a+b x\right ]-16 a^5 b x+120 a^5 b x \text {Log}\left [a+b x\right ]-68 a^4 b^2 x^2+60 a^4 b^2 x^2 \text {Log}\left [a+b x\right ]-20 a^3 b^3 x^3+5 a^2 b^4 x^4-2 a b^5 x^5+b^6 x^6}{4 b^7 \left (a^2+2 a b x+b^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^6/(a + b*x)^3,x]')

[Out]

(22 a ^ 6 + 60 a ^ 6 Log[a + b x] - 16 a ^ 5 b x + 120 a ^ 5 b x Log[a + b x] - 68 a ^ 4 b ^ 2 x ^ 2 + 60 a ^
4 b ^ 2 x ^ 2 Log[a + b x] - 20 a ^ 3 b ^ 3 x ^ 3 + 5 a ^ 2 b ^ 4 x ^ 4 - 2 a b ^ 5 x ^ 5 + b ^ 6 x ^ 6) / (4
b ^ 7 (a ^ 2 + 2 a b x + b ^ 2 x ^ 2))

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Maple [A]
time = 0.08, size = 83, normalized size = 0.97

method	result	size
risch	\(\frac {x^{4}}{4 b^{3}}-\frac {a \,x^{3}}{b^{4}}+\frac {3 a^{2} x^{2}}{b^{5}}-\frac {10 a^{3} x}{b^{6}}+\frac {6 a^{5} x +\frac {11 a^{6}}{2 b}}{b^{6} \left (b x +a \right )^{2}}+\frac {15 a^{4} \ln \left (b x +a \right )}{b^{7}}\)	\(79\)
norman	\(\frac {\frac {x^{6}}{4 b}-\frac {a \,x^{5}}{2 b^{2}}+\frac {45 a^{6}}{2 b^{7}}-\frac {5 a^{3} x^{3}}{b^{4}}+\frac {5 a^{2} x^{4}}{4 b^{3}}+\frac {30 a^{5} x}{b^{6}}}{\left (b x +a \right )^{2}}+\frac {15 a^{4} \ln \left (b x +a \right )}{b^{7}}\)	\(81\)
default	\(-\frac {-\frac {1}{4} b^{3} x^{4}+a \,b^{2} x^{3}-3 a^{2} b \,x^{2}+10 a^{3} x}{b^{6}}+\frac {6 a^{5}}{b^{7} \left (b x +a \right )}-\frac {a^{6}}{2 b^{7} \left (b x +a \right )^{2}}+\frac {15 a^{4} \ln \left (b x +a \right )}{b^{7}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/b^6*(-1/4*b^3*x^4+a*b^2*x^3-3*a^2*b*x^2+10*a^3*x)+6*a^5/b^7/(b*x+a)-1/2*a^6/b^7/(b*x+a)^2+15*a^4*ln(b*x+a)/
b^7

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Maxima [A]
time = 0.27, size = 91, normalized size = 1.06 \begin {gather*} \frac {12 \, a^{5} b x + 11 \, a^{6}}{2 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac {15 \, a^{4} \log \left (b x + a\right )}{b^{7}} + \frac {b^{3} x^{4} - 4 \, a b^{2} x^{3} + 12 \, a^{2} b x^{2} - 40 \, a^{3} x}{4 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^3,x, algorithm="maxima")

[Out]

1/2*(12*a^5*b*x + 11*a^6)/(b^9*x^2 + 2*a*b^8*x + a^2*b^7) + 15*a^4*log(b*x + a)/b^7 + 1/4*(b^3*x^4 - 4*a*b^2*x
^3 + 12*a^2*b*x^2 - 40*a^3*x)/b^6

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Fricas [A]
time = 0.30, size = 117, normalized size = 1.36 \begin {gather*} \frac {b^{6} x^{6} - 2 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{4} b^{2} x^{2} - 16 \, a^{5} b x + 22 \, a^{6} + 60 \, {\left (a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{4 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*x^6 - 2*a*b^5*x^5 + 5*a^2*b^4*x^4 - 20*a^3*b^3*x^3 - 68*a^4*b^2*x^2 - 16*a^5*b*x + 22*a^6 + 60*(a^4*b
^2*x^2 + 2*a^5*b*x + a^6)*log(b*x + a))/(b^9*x^2 + 2*a*b^8*x + a^2*b^7)

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Sympy [A]
time = 0.17, size = 92, normalized size = 1.07 \begin {gather*} \frac {15 a^{4} \log {\left (a + b x \right )}}{b^{7}} - \frac {10 a^{3} x}{b^{6}} + \frac {3 a^{2} x^{2}}{b^{5}} - \frac {a x^{3}}{b^{4}} + \frac {11 a^{6} + 12 a^{5} b x}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac {x^{4}}{4 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**3,x)

[Out]

15*a**4*log(a + b*x)/b**7 - 10*a**3*x/b**6 + 3*a**2*x**2/b**5 - a*x**3/b**4 + (11*a**6 + 12*a**5*b*x)/(2*a**2*
b**7 + 4*a*b**8*x + 2*b**9*x**2) + x**4/(4*b**3)

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Giac [A]
time = 0.00, size = 92, normalized size = 1.07 \begin {gather*} \frac {\frac {1}{4} x^{4} b^{9}-x^{3} b^{8} a+3 x^{2} b^{7} a^{2}-10 x b^{6} a^{3}}{b^{12}}+\frac {\frac {1}{2} \left (12 b a^{5} x+11 a^{6}\right )}{b^{7} \left (x b+a\right )^{2}}+\frac {15 a^{4} \ln \left |x b+a\right |}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^3,x)

[Out]

15*a^4*log(abs(b*x + a))/b^7 + 1/2*(12*a^5*b*x + 11*a^6)/((b*x + a)^2*b^7) + 1/4*(b^9*x^4 - 4*a*b^8*x^3 + 12*a
^2*b^7*x^2 - 40*a^3*b^6*x)/b^12

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Mupad [B]
time = 0.16, size = 78, normalized size = 0.91 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^4}{4}-2\,a\,{\left (a+b\,x\right )}^3+\frac {15\,a^2\,{\left (a+b\,x\right )}^2}{2}+\frac {6\,a^5}{a+b\,x}-\frac {a^6}{2\,{\left (a+b\,x\right )}^2}+15\,a^4\,\ln \left (a+b\,x\right )-20\,a^3\,b\,x}{b^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b*x)^3,x)

[Out]

((a + b*x)^4/4 - 2*a*(a + b*x)^3 + (15*a^2*(a + b*x)^2)/2 + (6*a^5)/(a + b*x) - a^6/(2*(a + b*x)^2) + 15*a^4*l
og(a + b*x) - 20*a^3*b*x)/b^7

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