3.1858 \(\int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx\)

Optimal. Leaf size=92 \[ -\frac{1215 (1-2 x)^{17/2}}{1088}+\frac{351}{20} (1-2 x)^{15/2}-\frac{97335}{832} (1-2 x)^{13/2}+\frac{37485}{88} (1-2 x)^{11/2}-\frac{173215}{192} (1-2 x)^{9/2}+\frac{8575}{8} (1-2 x)^{7/2}-\frac{184877}{320} (1-2 x)^{5/2} \]

[Out]

(-184877*(1 - 2*x)^(5/2))/320 + (8575*(1 - 2*x)^(7/2))/8 - (173215*(1 - 2*x)^(9/2))/192 + (37485*(1 - 2*x)^(11
/2))/88 - (97335*(1 - 2*x)^(13/2))/832 + (351*(1 - 2*x)^(15/2))/20 - (1215*(1 - 2*x)^(17/2))/1088

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Rubi [A]  time = 0.0162481, antiderivative size = 92, normalized size of antiderivative = 1., 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 = {77} \[ -\frac{1215 (1-2 x)^{17/2}}{1088}+\frac{351}{20} (1-2 x)^{15/2}-\frac{97335}{832} (1-2 x)^{13/2}+\frac{37485}{88} (1-2 x)^{11/2}-\frac{173215}{192} (1-2 x)^{9/2}+\frac{8575}{8} (1-2 x)^{7/2}-\frac{184877}{320} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-184877*(1 - 2*x)^(5/2))/320 + (8575*(1 - 2*x)^(7/2))/8 - (173215*(1 - 2*x)^(9/2))/192 + (37485*(1 - 2*x)^(11
/2))/88 - (97335*(1 - 2*x)^(13/2))/832 + (351*(1 - 2*x)^(15/2))/20 - (1215*(1 - 2*x)^(17/2))/1088

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx &=\int \left (\frac{184877}{64} (1-2 x)^{3/2}-\frac{60025}{8} (1-2 x)^{5/2}+\frac{519645}{64} (1-2 x)^{7/2}-\frac{37485}{8} (1-2 x)^{9/2}+\frac{97335}{64} (1-2 x)^{11/2}-\frac{1053}{4} (1-2 x)^{13/2}+\frac{1215}{64} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac{184877}{320} (1-2 x)^{5/2}+\frac{8575}{8} (1-2 x)^{7/2}-\frac{173215}{192} (1-2 x)^{9/2}+\frac{37485}{88} (1-2 x)^{11/2}-\frac{97335}{832} (1-2 x)^{13/2}+\frac{351}{20} (1-2 x)^{15/2}-\frac{1215 (1-2 x)^{17/2}}{1088}\\ \end{align*}

Mathematica [A]  time = 0.0188397, size = 43, normalized size = 0.47 \[ -\frac{(1-2 x)^{5/2} \left (2606175 x^6+12660219 x^5+26832465 x^4+32431860 x^3+24424220 x^2+11562520 x+3012632\right )}{36465} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(3012632 + 11562520*x + 24424220*x^2 + 32431860*x^3 + 26832465*x^4 + 12660219*x^5 + 2606175*
x^6))/36465

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{2606175\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+3012632}{36465} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36465*(2606175*x^6+12660219*x^5+26832465*x^4+32431860*x^3+24424220*x^2+11562520*x+3012632)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.2405, size = 86, normalized size = 0.93 \begin{align*} -\frac{1215}{1088} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} + \frac{351}{20} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{97335}{832} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{37485}{88} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{173215}{192} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{8575}{8} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{184877}{320} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215/1088*(-2*x + 1)^(17/2) + 351/20*(-2*x + 1)^(15/2) - 97335/832*(-2*x + 1)^(13/2) + 37485/88*(-2*x + 1)^(1
1/2) - 173215/192*(-2*x + 1)^(9/2) + 8575/8*(-2*x + 1)^(7/2) - 184877/320*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.31894, size = 201, normalized size = 2.18 \begin{align*} -\frac{1}{36465} \,{\left (10424700 \, x^{8} + 40216176 \, x^{7} + 59295159 \, x^{6} + 35057799 \, x^{5} - 5198095 \, x^{4} - 19014940 \, x^{3} - 9775332 \, x^{2} - 488008 \, x + 3012632\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36465*(10424700*x^8 + 40216176*x^7 + 59295159*x^6 + 35057799*x^5 - 5198095*x^4 - 19014940*x^3 - 9775332*x^2
- 488008*x + 3012632)*sqrt(-2*x + 1)

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Sympy [A]  time = 15.7555, size = 82, normalized size = 0.89 \begin{align*} - \frac{1215 \left (1 - 2 x\right )^{\frac{17}{2}}}{1088} + \frac{351 \left (1 - 2 x\right )^{\frac{15}{2}}}{20} - \frac{97335 \left (1 - 2 x\right )^{\frac{13}{2}}}{832} + \frac{37485 \left (1 - 2 x\right )^{\frac{11}{2}}}{88} - \frac{173215 \left (1 - 2 x\right )^{\frac{9}{2}}}{192} + \frac{8575 \left (1 - 2 x\right )^{\frac{7}{2}}}{8} - \frac{184877 \left (1 - 2 x\right )^{\frac{5}{2}}}{320} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215*(1 - 2*x)**(17/2)/1088 + 351*(1 - 2*x)**(15/2)/20 - 97335*(1 - 2*x)**(13/2)/832 + 37485*(1 - 2*x)**(11/2
)/88 - 173215*(1 - 2*x)**(9/2)/192 + 8575*(1 - 2*x)**(7/2)/8 - 184877*(1 - 2*x)**(5/2)/320

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Giac [A]  time = 2.5501, size = 153, normalized size = 1.66 \begin{align*} -\frac{1215}{1088} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} - \frac{351}{20} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{97335}{832} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{37485}{88} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{173215}{192} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{8575}{8} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{184877}{320} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215/1088*(2*x - 1)^8*sqrt(-2*x + 1) - 351/20*(2*x - 1)^7*sqrt(-2*x + 1) - 97335/832*(2*x - 1)^6*sqrt(-2*x +
1) - 37485/88*(2*x - 1)^5*sqrt(-2*x + 1) - 173215/192*(2*x - 1)^4*sqrt(-2*x + 1) - 8575/8*(2*x - 1)^3*sqrt(-2*
x + 1) - 184877/320*(2*x - 1)^2*sqrt(-2*x + 1)