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

Optimal. Leaf size=79 \[ \frac {675}{224} (1-2 x)^{7/2}-\frac {1539}{32} (1-2 x)^{5/2}+\frac {5847}{16} (1-2 x)^{3/2}-\frac {39977}{16} \sqrt {1-2 x}-\frac {91091}{32 \sqrt {1-2 x}}+\frac {41503}{96 (1-2 x)^{3/2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 79, 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 {675}{224} (1-2 x)^{7/2}-\frac {1539}{32} (1-2 x)^{5/2}+\frac {5847}{16} (1-2 x)^{3/2}-\frac {39977}{16} \sqrt {1-2 x}-\frac {91091}{32 \sqrt {1-2 x}}+\frac {41503}{96 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

41503/(96*(1 - 2*x)^(3/2)) - 91091/(32*Sqrt[1 - 2*x]) - (39977*Sqrt[1 - 2*x])/16 + (5847*(1 - 2*x)^(3/2))/16 -
(1539*(1 - 2*x)^(5/2))/32 + (675*(1 - 2*x)^(7/2))/224

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)^3 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {41503}{32 (1-2 x)^{5/2}}-\frac {91091}{32 (1-2 x)^{3/2}}+\frac {39977}{16 \sqrt {1-2 x}}-\frac {17541}{16} \sqrt {1-2 x}+\frac {7695}{32} (1-2 x)^{3/2}-\frac {675}{32} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac {41503}{96 (1-2 x)^{3/2}}-\frac {91091}{32 \sqrt {1-2 x}}-\frac {39977}{16} \sqrt {1-2 x}+\frac {5847}{16} (1-2 x)^{3/2}-\frac {1539}{32} (1-2 x)^{5/2}+\frac {675}{224} (1-2 x)^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {2025 x^5+11097 x^4+34137 x^3+139497 x^2-290838 x+96442}{21 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(96442 - 290838*x + 139497*x^2 + 34137*x^3 + 11097*x^4 + 2025*x^5)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.03, size = 58, normalized size = 0.73 \begin {gather*} \frac {2025 (1-2 x)^5-32319 (1-2 x)^4+245574 (1-2 x)^3-1679034 (1-2 x)^2-1912911 (1-2 x)+290521}{672 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(290521 - 1912911*(1 - 2*x) - 1679034*(1 - 2*x)^2 + 245574*(1 - 2*x)^3 - 32319*(1 - 2*x)^4 + 2025*(1 - 2*x)^5)
/(672*(1 - 2*x)^(3/2))

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fricas [A]  time = 0.89, size = 46, normalized size = 0.58 \begin {gather*} -\frac {{\left (2025 \, x^{5} + 11097 \, x^{4} + 34137 \, x^{3} + 139497 \, x^{2} - 290838 \, x + 96442\right )} \sqrt {-2 \, x + 1}}{21 \, {\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)^3*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/21*(2025*x^5 + 11097*x^4 + 34137*x^3 + 139497*x^2 - 290838*x + 96442)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.25, size = 72, normalized size = 0.91 \begin {gather*} -\frac {675}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1539}{32} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {5847}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {39977}{16} \, \sqrt {-2 \, x + 1} - \frac {539 \, {\left (507 \, x - 215\right )}}{48 \, {\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)^3*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-675/224*(2*x - 1)^3*sqrt(-2*x + 1) - 1539/32*(2*x - 1)^2*sqrt(-2*x + 1) + 5847/16*(-2*x + 1)^(3/2) - 39977/16
*sqrt(-2*x + 1) - 539/48*(507*x - 215)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {2025 x^{5}+11097 x^{4}+34137 x^{3}+139497 x^{2}-290838 x +96442}{21 \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)^3*(5*x+3)^2/(-2*x+1)^(5/2),x)

[Out]

-1/21*(2025*x^5+11097*x^4+34137*x^3+139497*x^2-290838*x+96442)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.46, size = 51, normalized size = 0.65 \begin {gather*} \frac {675}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1539}{32} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {5847}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {39977}{16} \, \sqrt {-2 \, x + 1} + \frac {539 \, {\left (507 \, x - 215\right )}}{48 \, {\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)^3*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

675/224*(-2*x + 1)^(7/2) - 1539/32*(-2*x + 1)^(5/2) + 5847/16*(-2*x + 1)^(3/2) - 39977/16*sqrt(-2*x + 1) + 539
/48*(507*x - 215)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 50, normalized size = 0.63 \begin {gather*} \frac {\frac {91091\,x}{16}-\frac {115885}{48}}{{\left (1-2\,x\right )}^{3/2}}-\frac {39977\,\sqrt {1-2\,x}}{16}+\frac {5847\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {1539\,{\left (1-2\,x\right )}^{5/2}}{32}+\frac {675\,{\left (1-2\,x\right )}^{7/2}}{224} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((91091*x)/16 - 115885/48)/(1 - 2*x)^(3/2) - (39977*(1 - 2*x)^(1/2))/16 + (5847*(1 - 2*x)^(3/2))/16 - (1539*(1
- 2*x)^(5/2))/32 + (675*(1 - 2*x)^(7/2))/224

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sympy [A]  time = 29.85, size = 70, normalized size = 0.89 \begin {gather*} \frac {675 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} - \frac {1539 \left (1 - 2 x\right )^{\frac {5}{2}}}{32} + \frac {5847 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} - \frac {39977 \sqrt {1 - 2 x}}{16} - \frac {91091}{32 \sqrt {1 - 2 x}} + \frac {41503}{96 \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)**3*(3+5*x)**2/(1-2*x)**(5/2),x)

[Out]

675*(1 - 2*x)**(7/2)/224 - 1539*(1 - 2*x)**(5/2)/32 + 5847*(1 - 2*x)**(3/2)/16 - 39977*sqrt(1 - 2*x)/16 - 9109
1/(32*sqrt(1 - 2*x)) + 41503/(96*(1 - 2*x)**(3/2))

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