3.2149 \(\int \frac{(2+3 x)^2 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{45}{16} (1-2 x)^{5/2}+\frac{85}{2} (1-2 x)^{3/2}-\frac{3467}{8} \sqrt{1-2 x}-\frac{1309}{2 \sqrt{1-2 x}}+\frac{5929}{48 (1-2 x)^{3/2}} \]

[Out]

5929/(48*(1 - 2*x)^(3/2)) - 1309/(2*Sqrt[1 - 2*x]) - (3467*Sqrt[1 - 2*x])/8 + (85*(1 - 2*x)^(3/2))/2 - (45*(1
- 2*x)^(5/2))/16

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Rubi [A]  time = 0.0134463, antiderivative size = 66, 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} \[ -\frac{45}{16} (1-2 x)^{5/2}+\frac{85}{2} (1-2 x)^{3/2}-\frac{3467}{8} \sqrt{1-2 x}-\frac{1309}{2 \sqrt{1-2 x}}+\frac{5929}{48 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5929/(48*(1 - 2*x)^(3/2)) - 1309/(2*Sqrt[1 - 2*x]) - (3467*Sqrt[1 - 2*x])/8 + (85*(1 - 2*x)^(3/2))/2 - (45*(1
- 2*x)^(5/2))/16

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)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{5929}{16 (1-2 x)^{5/2}}-\frac{1309}{2 (1-2 x)^{3/2}}+\frac{3467}{8 \sqrt{1-2 x}}-\frac{255}{2} \sqrt{1-2 x}+\frac{225}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac{5929}{48 (1-2 x)^{3/2}}-\frac{1309}{2 \sqrt{1-2 x}}-\frac{3467}{8} \sqrt{1-2 x}+\frac{85}{2} (1-2 x)^{3/2}-\frac{45}{16} (1-2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0131909, size = 33, normalized size = 0.5 \[ -\frac{135 x^4+750 x^3+3873 x^2-8430 x+2774}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2774 - 8430*x + 3873*x^2 + 750*x^3 + 135*x^4)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.004, size = 30, normalized size = 0.5 \begin{align*} -{\frac{135\,{x}^{4}+750\,{x}^{3}+3873\,{x}^{2}-8430\,x+2774}{3} \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)^2/(1-2*x)^(5/2),x)

[Out]

-1/3*(135*x^4+750*x^3+3873*x^2-8430*x+2774)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.70428, size = 57, normalized size = 0.86 \begin{align*} -\frac{45}{16} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{85}{2} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{3467}{8} \, \sqrt{-2 \, x + 1} + \frac{77 \,{\left (816 \, x - 331\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)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-45/16*(-2*x + 1)^(5/2) + 85/2*(-2*x + 1)^(3/2) - 3467/8*sqrt(-2*x + 1) + 77/48*(816*x - 331)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.53527, size = 116, normalized size = 1.76 \begin{align*} -\frac{{\left (135 \, x^{4} + 750 \, x^{3} + 3873 \, x^{2} - 8430 \, x + 2774\right )} \sqrt{-2 \, x + 1}}{3 \,{\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)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(135*x^4 + 750*x^3 + 3873*x^2 - 8430*x + 2774)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 15.1194, size = 58, normalized size = 0.88 \begin{align*} - \frac{45 \left (1 - 2 x\right )^{\frac{5}{2}}}{16} + \frac{85 \left (1 - 2 x\right )^{\frac{3}{2}}}{2} - \frac{3467 \sqrt{1 - 2 x}}{8} - \frac{1309}{2 \sqrt{1 - 2 x}} + \frac{5929}{48 \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)**2/(1-2*x)**(5/2),x)

[Out]

-45*(1 - 2*x)**(5/2)/16 + 85*(1 - 2*x)**(3/2)/2 - 3467*sqrt(1 - 2*x)/8 - 1309/(2*sqrt(1 - 2*x)) + 5929/(48*(1
- 2*x)**(3/2))

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Giac [A]  time = 2.45791, size = 76, normalized size = 1.15 \begin{align*} -\frac{45}{16} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{85}{2} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{3467}{8} \, \sqrt{-2 \, x + 1} - \frac{77 \,{\left (816 \, x - 331\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)^2/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-45/16*(2*x - 1)^2*sqrt(-2*x + 1) + 85/2*(-2*x + 1)^(3/2) - 3467/8*sqrt(-2*x + 1) - 77/48*(816*x - 331)/((2*x
- 1)*sqrt(-2*x + 1))