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

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

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Rubi [A]  time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.47 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 82, normalized size = 0.89 \begin {gather*} \frac {-13030875 (1-2 x)^{17/2}[phone] (1-2 x)^{15/2}-1338919065 (1-2 x)^{13/2}[phone] (1-2 x)^{11/2}-10124519405 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-6356308959 (1-2 x)^{5/2}}{7001280} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6356308959*(1 - 2*x)^(5/2) + 11910708810*(1 - 2*x)^(7/2) - 10124519405*(1 - 2*x)^(9/2) + 4827661020*(1 - 2*x
)^(11/2) - 1338919065*(1 - 2*x)^(13/2) + 202818330*(1 - 2*x)^(15/2) - 13030875*(1 - 2*x)^(17/2))/7001280

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fricas [A]  time = 1.38, size = 49, normalized size = 0.53 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.89, size = 113, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.01, size = 40, normalized size = 0.43 \begin {gather*} -\frac {\left (13030875 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{109395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(3*x+2)^4*(5*x+3)^2,x)

[Out]

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

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maxima [A]  time = 0.56, size = 64, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {54439\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {290521\,{\left (1-2\,x\right )}^{5/2}}{320}-\frac {832951\,{\left (1-2\,x\right )}^{9/2}}{576}+\frac {121359\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {159111\,{\left (1-2\,x\right )}^{13/2}}{832}+\frac {927\,{\left (1-2\,x\right )}^{15/2}}{32}-\frac {2025\,{\left (1-2\,x\right )}^{17/2}}{1088} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)*(3*x + 2)^4*(5*x + 3)^2,x)

[Out]

(54439*(1 - 2*x)^(7/2))/32 - (290521*(1 - 2*x)^(5/2))/320 - (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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sympy [A]  time = 21.12, size = 82, normalized size = 0.89 \begin {gather*} - \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 {gather*}

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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