3.2160 \(\int \frac{(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{1125}{224} (1-2 x)^{7/2}-\frac{2535}{32} (1-2 x)^{5/2}+\frac{28555}{48} (1-2 x)^{3/2}-\frac{64317}{16} \sqrt{1-2 x}-\frac{144837}{32 \sqrt{1-2 x}}+\frac{65219}{96 (1-2 x)^{3/2}} \]

[Out]

65219/(96*(1 - 2*x)^(3/2)) - 144837/(32*Sqrt[1 - 2*x]) - (64317*Sqrt[1 - 2*x])/16 + (28555*(1 - 2*x)^(3/2))/48
- (2535*(1 - 2*x)^(5/2))/32 + (1125*(1 - 2*x)^(7/2))/224

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Rubi [A]  time = 0.0148918, antiderivative size = 79, 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} \[ \frac{1125}{224} (1-2 x)^{7/2}-\frac{2535}{32} (1-2 x)^{5/2}+\frac{28555}{48} (1-2 x)^{3/2}-\frac{64317}{16} \sqrt{1-2 x}-\frac{144837}{32 \sqrt{1-2 x}}+\frac{65219}{96 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

65219/(96*(1 - 2*x)^(3/2)) - 144837/(32*Sqrt[1 - 2*x]) - (64317*Sqrt[1 - 2*x])/16 + (28555*(1 - 2*x)^(3/2))/48
- (2535*(1 - 2*x)^(5/2))/32 + (1125*(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)^2 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{65219}{32 (1-2 x)^{5/2}}-\frac{144837}{32 (1-2 x)^{3/2}}+\frac{64317}{16 \sqrt{1-2 x}}-\frac{28555}{16} \sqrt{1-2 x}+\frac{12675}{32} (1-2 x)^{3/2}-\frac{1125}{32} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac{65219}{96 (1-2 x)^{3/2}}-\frac{144837}{32 \sqrt{1-2 x}}-\frac{64317}{16} \sqrt{1-2 x}+\frac{28555}{48} (1-2 x)^{3/2}-\frac{2535}{32} (1-2 x)^{5/2}+\frac{1125}{224} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.015261, size = 38, normalized size = 0.48 \[ -\frac{3375 x^5+18180 x^4+55145 x^3+223458 x^2-465060 x+154264}{21 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(154264 - 465060*x + 223458*x^2 + 55145*x^3 + 18180*x^4 + 3375*x^5)/(21*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{3375\,{x}^{5}+18180\,{x}^{4}+55145\,{x}^{3}+223458\,{x}^{2}-465060\,x+154264}{21} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(3375*x^5+18180*x^4+55145*x^3+223458*x^2-465060*x+154264)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.03948, size = 69, normalized size = 0.87 \begin{align*} \frac{1125}{224} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2535}{32} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{28555}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{64317}{16} \, \sqrt{-2 \, x + 1} + \frac{847 \,{\left (513 \, x - 218\right )}}{48 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/224*(-2*x + 1)^(7/2) - 2535/32*(-2*x + 1)^(5/2) + 28555/48*(-2*x + 1)^(3/2) - 64317/16*sqrt(-2*x + 1) + 8
47/48*(513*x - 218)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.54942, size = 146, normalized size = 1.85 \begin{align*} -\frac{{\left (3375 \, x^{5} + 18180 \, x^{4} + 55145 \, x^{3} + 223458 \, x^{2} - 465060 \, x + 154264\right )} \sqrt{-2 \, x + 1}}{21 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(3375*x^5 + 18180*x^4 + 55145*x^3 + 223458*x^2 - 465060*x + 154264)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 19.4738, size = 70, normalized size = 0.89 \begin{align*} \frac{1125 \left (1 - 2 x\right )^{\frac{7}{2}}}{224} - \frac{2535 \left (1 - 2 x\right )^{\frac{5}{2}}}{32} + \frac{28555 \left (1 - 2 x\right )^{\frac{3}{2}}}{48} - \frac{64317 \sqrt{1 - 2 x}}{16} - \frac{144837}{32 \sqrt{1 - 2 x}} + \frac{65219}{96 \left (1 - 2 x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125*(1 - 2*x)**(7/2)/224 - 2535*(1 - 2*x)**(5/2)/32 + 28555*(1 - 2*x)**(3/2)/48 - 64317*sqrt(1 - 2*x)/16 - 14
4837/(32*sqrt(1 - 2*x)) + 65219/(96*(1 - 2*x)**(3/2))

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Giac [A]  time = 2.2756, size = 97, normalized size = 1.23 \begin{align*} -\frac{1125}{224} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2535}{32} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{28555}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{64317}{16} \, \sqrt{-2 \, x + 1} - \frac{847 \,{\left (513 \, x - 218\right )}}{48 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1125/224*(2*x - 1)^3*sqrt(-2*x + 1) - 2535/32*(2*x - 1)^2*sqrt(-2*x + 1) + 28555/48*(-2*x + 1)^(3/2) - 64317/
16*sqrt(-2*x + 1) - 847/48*(513*x - 218)/((2*x - 1)*sqrt(-2*x + 1))