∫ (3+5x)3 ______________________

3.14.43 \(\int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx\)

Optimal. Leaf size=87 \[ -\frac {5324}{117649 (3 x+2)}-\frac {1331}{16807 (3 x+2)^2}-\frac {1331}{7203 (3 x+2)^3}+\frac {3469}{37044 (3 x+2)^4}-\frac {103}{6615 (3 x+2)^5}+\frac {1}{1134 (3 x+2)^6}-\frac {10648 \log (1-2 x)}{823543}+\frac {10648 \log (3 x+2)}{823543} \]

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Rubi [A] time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} -\frac {5324}{117649 (3 x+2)}-\frac {1331}{16807 (3 x+2)^2}-\frac {1331}{7203 (3 x+2)^3}+\frac {3469}{37044 (3 x+2)^4}-\frac {103}{6615 (3 x+2)^5}+\frac {1}{1134 (3 x+2)^6}-\frac {10648 \log (1-2 x)}{823543}+\frac {10648 \log (3 x+2)}{823543} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

1/(1134*(2 + 3*x)^6) - 103/(6615*(2 + 3*x)^5) + 3469/(37044*(2 + 3*x)^4) - 1331/(7203*(2 + 3*x)^3) - 1331/(168

07*(2 + 3*x)^2) - 5324/(117649*(2 + 3*x)) - (10648*Log[1 - 2*x])/823543 + (10648*Log[2 + 3*x])/823543

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI

ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&

(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx &=\int \left (-\frac {21296}{823543 (-1+2 x)}-\frac {1}{63 (2+3 x)^7}+\frac {103}{441 (2+3 x)^6}-\frac {3469}{3087 (2+3 x)^5}+\frac {3993}{2401 (2+3 x)^4}+\frac {7986}{16807 (2+3 x)^3}+\frac {15972}{117649 (2+3 x)^2}+\frac {31944}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {1}{1134 (2+3 x)^6}-\frac {103}{6615 (2+3 x)^5}+\frac {3469}{37044 (2+3 x)^4}-\frac {1331}{7203 (2+3 x)^3}-\frac {1331}{16807 (2+3 x)^2}-\frac {5324}{117649 (2+3 x)}-\frac {10648 \log (1-2 x)}{823543}+\frac {10648 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A] time = 0.08, size = 57, normalized size = 0.66 \begin {gather*} \frac {4 \left (-\frac {7 \left (2095845840 x^5+8208729540 x^4+13525968060 x^3+11211272235 x^2+4581535248 x+733614062\right )}{16 (3 x+2)^6}-1078110 \log (1-2 x)+1078110 \log (6 x+4)\right )}{333534915} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

(4*((-7*(733614062 + 4581535248*x + 11211272235*x^2 + 13525968060*x^3 + 8208729540*x^4 + 2095845840*x^5))/(16*

(2 + 3*x)^6) - 1078110*Log[1 - 2*x] + 1078110*Log[4 + 6*x]))/333534915

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^7), x]

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fricas [A] time = 1.54, size = 135, normalized size = 1.55 \begin {gather*} -\frac {14670920880 \, x^{5} + 57461106780 \, x^{4} + 94681776420 \, x^{3} + 78478905645 \, x^{2} - 17249760 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 17249760 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (2 \, x - 1\right ) + 32070746736 \, x + 5135298434}{1334139660 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/1334139660*(14670920880*x^5 + 57461106780*x^4 + 94681776420*x^3 + 78478905645*x^2 - 17249760*(729*x^6 + 291

6*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 17249760*(729*x^6 + 2916*x^5 + 4860*x^4 +

4320*x^3 + 2160*x^2 + 576*x + 64)*log(2*x - 1) + 32070746736*x + 5135298434)/(729*x^6 + 2916*x^5 + 4860*x^4 +

4320*x^3 + 2160*x^2 + 576*x + 64)

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giac [A] time = 0.85, size = 53, normalized size = 0.61 \begin {gather*} -\frac {2095845840 \, x^{5} + 8208729540 \, x^{4} + 13525968060 \, x^{3} + 11211272235 \, x^{2} + 4581535248 \, x + 733614062}{190591380 \, {\left (3 \, x + 2\right )}^{6}} + \frac {10648}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {10648}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/190591380*(2095845840*x^5 + 8208729540*x^4 + 13525968060*x^3 + 11211272235*x^2 + 4581535248*x + 733614062)/

(3*x + 2)^6 + 10648/823543*log(abs(3*x + 2)) - 10648/823543*log(abs(2*x - 1))

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maple [A] time = 0.01, size = 72, normalized size = 0.83 \begin {gather*} -\frac {10648 \ln \left (2 x -1\right )}{823543}+\frac {10648 \ln \left (3 x +2\right )}{823543}+\frac {1}{1134 \left (3 x +2\right )^{6}}-\frac {103}{6615 \left (3 x +2\right )^{5}}+\frac {3469}{37044 \left (3 x +2\right )^{4}}-\frac {1331}{7203 \left (3 x +2\right )^{3}}-\frac {1331}{16807 \left (3 x +2\right )^{2}}-\frac {5324}{117649 \left (3 x +2\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(1-2*x)/(3*x+2)^7,x)

[Out]

1/1134/(3*x+2)^6-103/6615/(3*x+2)^5+3469/37044/(3*x+2)^4-1331/7203/(3*x+2)^3-1331/16807/(3*x+2)^2-5324/117649/

(3*x+2)+10648/823543*ln(3*x+2)-10648/823543*ln(2*x-1)

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maxima [A] time = 0.57, size = 76, normalized size = 0.87 \begin {gather*} -\frac {2095845840 \, x^{5} + 8208729540 \, x^{4} + 13525968060 \, x^{3} + 11211272235 \, x^{2} + 4581535248 \, x + 733614062}{190591380 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {10648}{823543} \, \log \left (3 \, x + 2\right ) - \frac {10648}{823543} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/190591380*(2095845840*x^5 + 8208729540*x^4 + 13525968060*x^3 + 11211272235*x^2 + 4581535248*x + 733614062)/

(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 10648/823543*log(3*x + 2) - 10648/823543*

log(2*x - 1)

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mupad [B] time = 1.12, size = 66, normalized size = 0.76 \begin {gather*} \frac {21296\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {5324\,x^5}{352947}+\frac {62557\,x^4}{1058841}+\frac {927707\,x^3}{9529569}+\frac {83046461\,x^2}{1029193452}+\frac {127264868\,x}{3859475445}+\frac {366807031}{69470558010}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)^3/((2*x - 1)*(3*x + 2)^7),x)

[Out]

(21296*atanh((12*x)/7 + 1/7))/823543 - ((127264868*x)/3859475445 + (83046461*x^2)/1029193452 + (927707*x^3)/95

29569 + (62557*x^4)/1058841 + (5324*x^5)/352947 + 366807031/69470558010)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/

27 + (20*x^4)/3 + 4*x^5 + x^6 + 64/729)

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sympy [A] time = 0.22, size = 75, normalized size = 0.86 \begin {gather*} - \frac {2095845840 x^{5} + 8208729540 x^{4} + 13525968060 x^{3} + 11211272235 x^{2} + 4581535248 x + 733614062}{138941116020 x^{6} + 555764464080 x^{5} + 926274106800 x^{4} + 823354761600 x^{3} + 411677380800 x^{2} + 109780634880 x + 12197848320} - \frac {10648 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {10648 \log {\left (x + \frac {2}{3} \right )}}{823543} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)/(2+3*x)**7,x)

[Out]

-(2095845840*x**5 + 8208729540*x**4 + 13525968060*x**3 + 11211272235*x**2 + 4581535248*x + 733614062)/(1389411

16020*x**6 + 555764464080*x**5 + 926274106800*x**4 + 823354761600*x**3 + 411677380800*x**2 + 109780634880*x +

12197848320) - 10648*log(x - 1/2)/823543 + 10648*log(x + 2/3)/823543

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