3.1870 \(\int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx\)

Optimal. Leaf size=92 \[ -\frac{2025 (1-2 x)^{17/2}}{1088}+\frac{927}{32} (1-2 x)^{15/2}-\frac{159111}{832} (1-2 x)^{13/2}+\frac{121359}{176} (1-2 x)^{11/2}-\frac{832951}{576} (1-2 x)^{9/2}+\frac{54439}{32} (1-2 x)^{7/2}-\frac{290521}{320} (1-2 x)^{5/2} \]

[Out]

(-290521*(1 - 2*x)^(5/2))/320 + (54439*(1 - 2*x)^(7/2))/32 - (832951*(1 - 2*x)^(9/2))/576 + (121359*(1 - 2*x)^
(11/2))/176 - (159111*(1 - 2*x)^(13/2))/832 + (927*(1 - 2*x)^(15/2))/32 - (2025*(1 - 2*x)^(17/2))/1088

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Rubi [A]  time = 0.0175019, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {88} \[ -\frac{2025 (1-2 x)^{17/2}}{1088}+\frac{927}{32} (1-2 x)^{15/2}-\frac{159111}{832} (1-2 x)^{13/2}+\frac{121359}{176} (1-2 x)^{11/2}-\frac{832951}{576} (1-2 x)^{9/2}+\frac{54439}{32} (1-2 x)^{7/2}-\frac{290521}{320} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^2,x]

[Out]

(-290521*(1 - 2*x)^(5/2))/320 + (54439*(1 - 2*x)^(7/2))/32 - (832951*(1 - 2*x)^(9/2))/576 + (121359*(1 - 2*x)^
(11/2))/176 - (159111*(1 - 2*x)^(13/2))/832 + (927*(1 - 2*x)^(15/2))/32 - (2025*(1 - 2*x)^(17/2))/1088

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 (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx &=\int \left (\frac{290521}{64} (1-2 x)^{3/2}-\frac{381073}{32} (1-2 x)^{5/2}+\frac{832951}{64} (1-2 x)^{7/2}-\frac{121359}{16} (1-2 x)^{9/2}+\frac{159111}{64} (1-2 x)^{11/2}-\frac{13905}{32} (1-2 x)^{13/2}+\frac{2025}{64} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac{290521}{320} (1-2 x)^{5/2}+\frac{54439}{32} (1-2 x)^{7/2}-\frac{832951}{576} (1-2 x)^{9/2}+\frac{121359}{176} (1-2 x)^{11/2}-\frac{159111}{832} (1-2 x)^{13/2}+\frac{927}{32} (1-2 x)^{15/2}-\frac{2025 (1-2 x)^{17/2}}{1088}\\ \end{align*}

Mathematica [A]  time = 0.0196016, size = 43, normalized size = 0.47 \[ -\frac{(1-2 x)^{5/2} \left (13030875 x^6+62316540 x^5+130072635 x^4+154943820 x^3+115145660 x^2+53902600 x+13931096\right )}{109395} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^2,x]

[Out]

-((1 - 2*x)^(5/2)*(13931096 + 53902600*x + 115145660*x^2 + 154943820*x^3 + 130072635*x^4 + 62316540*x^5 + 1303
0875*x^6))/109395

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{13030875\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+13931096}{109395} \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)^4*(3+5*x)^2,x)

[Out]

-1/109395*(13030875*x^6+62316540*x^5+130072635*x^4+154943820*x^3+115145660*x^2+53902600*x+13931096)*(1-2*x)^(5
/2)

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Maxima [A]  time = 1.06303, size = 86, normalized size = 0.93 \begin{align*} -\frac{2025}{1088} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} + \frac{927}{32} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{159111}{832} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{121359}{176} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{832951}{576} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{54439}{32} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{290521}{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)^4*(3+5*x)^2,x, algorithm="maxima")

[Out]

-2025/1088*(-2*x + 1)^(17/2) + 927/32*(-2*x + 1)^(15/2) - 159111/832*(-2*x + 1)^(13/2) + 121359/176*(-2*x + 1)
^(11/2) - 832951/576*(-2*x + 1)^(9/2) + 54439/32*(-2*x + 1)^(7/2) - 290521/320*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.36156, size = 212, normalized size = 2.3 \begin{align*} -\frac{1}{109395} \,{\left (52123500 \, x^{8} + 197142660 \, x^{7} + 284055255 \, x^{6} + 161801280 \, x^{5} - 29120005 \, x^{4} - 90028420 \, x^{3} - 44740356 \, x^{2} - 1821784 \, x + 13931096\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)^4*(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/109395*(52123500*x^8 + 197142660*x^7 + 284055255*x^6 + 161801280*x^5 - 29120005*x^4 - 90028420*x^3 - 447403
56*x^2 - 1821784*x + 13931096)*sqrt(-2*x + 1)

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Sympy [A]  time = 15.9024, size = 82, normalized size = 0.89 \begin{align*} - \frac{2025 \left (1 - 2 x\right )^{\frac{17}{2}}}{1088} + \frac{927 \left (1 - 2 x\right )^{\frac{15}{2}}}{32} - \frac{159111 \left (1 - 2 x\right )^{\frac{13}{2}}}{832} + \frac{121359 \left (1 - 2 x\right )^{\frac{11}{2}}}{176} - \frac{832951 \left (1 - 2 x\right )^{\frac{9}{2}}}{576} + \frac{54439 \left (1 - 2 x\right )^{\frac{7}{2}}}{32} - \frac{290521 \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)**4*(3+5*x)**2,x)

[Out]

-2025*(1 - 2*x)**(17/2)/1088 + 927*(1 - 2*x)**(15/2)/32 - 159111*(1 - 2*x)**(13/2)/832 + 121359*(1 - 2*x)**(11
/2)/176 - 832951*(1 - 2*x)**(9/2)/576 + 54439*(1 - 2*x)**(7/2)/32 - 290521*(1 - 2*x)**(5/2)/320

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Giac [A]  time = 1.71029, size = 153, normalized size = 1.66 \begin{align*} -\frac{2025}{1088} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} - \frac{927}{32} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{159111}{832} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{121359}{176} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{832951}{576} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{54439}{32} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{290521}{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)^4*(3+5*x)^2,x, algorithm="giac")

[Out]

-2025/1088*(2*x - 1)^8*sqrt(-2*x + 1) - 927/32*(2*x - 1)^7*sqrt(-2*x + 1) - 159111/832*(2*x - 1)^6*sqrt(-2*x +
1) - 121359/176*(2*x - 1)^5*sqrt(-2*x + 1) - 832951/576*(2*x - 1)^4*sqrt(-2*x + 1) - 54439/32*(2*x - 1)^3*sqr
t(-2*x + 1) - 290521/320*(2*x - 1)^2*sqrt(-2*x + 1)