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

Optimal. Leaf size=92 \[ -\frac {225}{64} (1-2 x)^{9/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {832951}{64} \sqrt {1-2 x}-\frac {381073}{32 \sqrt {1-2 x}}+\frac {290521}{192 (1-2 x)^{3/2}} \]

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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 {225}{64} (1-2 x)^{9/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {832951}{64} \sqrt {1-2 x}-\frac {381073}{32 \sqrt {1-2 x}}+\frac {290521}{192 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

290521/(192*(1 - 2*x)^(3/2)) - 381073/(32*Sqrt[1 - 2*x]) - (832951*Sqrt[1 - 2*x])/64 + (40453*(1 - 2*x)^(3/2))
/16 - (159111*(1 - 2*x)^(5/2))/320 + (13905*(1 - 2*x)^(7/2))/224 - (225*(1 - 2*x)^(9/2))/64

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 {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {290521}{64 (1-2 x)^{5/2}}-\frac {381073}{32 (1-2 x)^{3/2}}+\frac {832951}{64 \sqrt {1-2 x}}-\frac {121359}{16} \sqrt {1-2 x}+\frac {159111}{64} (1-2 x)^{3/2}-\frac {13905}{32} (1-2 x)^{5/2}+\frac {2025}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {290521}{192 (1-2 x)^{3/2}}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951}{64} \sqrt {1-2 x}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {225}{64} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {23625 x^6+137700 x^5+402489 x^4+915492 x^3+3294996 x^2-6731112 x+2238664}{105 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(2238664 - 6731112*x + 3294996*x^2 + 915492*x^3 + 402489*x^4 + 137700*x^5 + 23625*x^6)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.03, size = 67, normalized size = 0.73 \begin {gather*} \frac {-23625 (1-2 x)^6+417150 (1-2 x)^5-3341331 (1-2 x)^4+16990260 (1-2 x)^3-87459855 (1-2 x)^2-80025330 (1-2 x)[phone]}{6720 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10168235 - 80025330*(1 - 2*x) - 87459855*(1 - 2*x)^2 + 16990260*(1 - 2*x)^3 - 3341331*(1 - 2*x)^4 + 417150*(1
- 2*x)^5 - 23625*(1 - 2*x)^6)/(6720*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.18, size = 51, normalized size = 0.55 \begin {gather*} -\frac {{\left (23625 \, x^{6} + 137700 \, x^{5} + 402489 \, x^{4} + 915492 \, x^{3} + 3294996 \, x^{2} - 6731112 \, x + 2238664\right )} \sqrt {-2 \, x + 1}}{105 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/105*(23625*x^6 + 137700*x^5 + 402489*x^4 + 915492*x^3 + 3294996*x^2 - 6731112*x + 2238664)*sqrt(-2*x + 1)/(
4*x^2 - 4*x + 1)

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giac [A]  time = 1.23, size = 88, normalized size = 0.96 \begin {gather*} -\frac {225}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {13905}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {159111}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {40453}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {832951}{64} \, \sqrt {-2 \, x + 1} - \frac {3773 \, {\left (1212 \, x - 529\right )}}{192 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-225/64*(2*x - 1)^4*sqrt(-2*x + 1) - 13905/224*(2*x - 1)^3*sqrt(-2*x + 1) - 159111/320*(2*x - 1)^2*sqrt(-2*x +
1) + 40453/16*(-2*x + 1)^(3/2) - 832951/64*sqrt(-2*x + 1) - 3773/192*(1212*x - 529)/((2*x - 1)*sqrt(-2*x + 1)
)

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \begin {gather*} -\frac {23625 x^{6}+137700 x^{5}+402489 x^{4}+915492 x^{3}+3294996 x^{2}-6731112 x +2238664}{105 \left (-2 x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(23625*x^6+137700*x^5+402489*x^4+915492*x^3+3294996*x^2-6731112*x+2238664)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.61, size = 60, normalized size = 0.65 \begin {gather*} -\frac {225}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {13905}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {159111}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {40453}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {832951}{64} \, \sqrt {-2 \, x + 1} + \frac {3773 \, {\left (1212 \, x - 529\right )}}{192 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-225/64*(-2*x + 1)^(9/2) + 13905/224*(-2*x + 1)^(7/2) - 159111/320*(-2*x + 1)^(5/2) + 40453/16*(-2*x + 1)^(3/2
) - 832951/64*sqrt(-2*x + 1) + 3773/192*(1212*x - 529)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 59, normalized size = 0.64 \begin {gather*} \frac {\frac {381073\,x}{16}-\frac {1995917}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {832951\,\sqrt {1-2\,x}}{64}+\frac {40453\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {159111\,{\left (1-2\,x\right )}^{5/2}}{320}+\frac {13905\,{\left (1-2\,x\right )}^{7/2}}{224}-\frac {225\,{\left (1-2\,x\right )}^{9/2}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((381073*x)/16 - 1995917/192)/(1 - 2*x)^(3/2) - (832951*(1 - 2*x)^(1/2))/64 + (40453*(1 - 2*x)^(3/2))/16 - (15
9111*(1 - 2*x)^(5/2))/320 + (13905*(1 - 2*x)^(7/2))/224 - (225*(1 - 2*x)^(9/2))/64

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sympy [A]  time = 36.47, size = 82, normalized size = 0.89 \begin {gather*} - \frac {225 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {13905 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} - \frac {159111 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} + \frac {40453 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} - \frac {832951 \sqrt {1 - 2 x}}{64} - \frac {381073}{32 \sqrt {1 - 2 x}} + \frac {290521}{192 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225*(1 - 2*x)**(9/2)/64 + 13905*(1 - 2*x)**(7/2)/224 - 159111*(1 - 2*x)**(5/2)/320 + 40453*(1 - 2*x)**(3/2)/1
6 - 832951*sqrt(1 - 2*x)/64 - 381073/(32*sqrt(1 - 2*x)) + 290521/(192*(1 - 2*x)**(3/2))

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