3.2086 \(\int \frac{(2+3 x)^2 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{225}{112} (1-2 x)^{7/2}+\frac{51}{2} (1-2 x)^{5/2}-\frac{3467}{24} (1-2 x)^{3/2}+\frac{1309}{2} \sqrt{1-2 x}+\frac{5929}{16 \sqrt{1-2 x}} \]

[Out]

5929/(16*Sqrt[1 - 2*x]) + (1309*Sqrt[1 - 2*x])/2 - (3467*(1 - 2*x)^(3/2))/24 + (51*(1 - 2*x)^(5/2))/2 - (225*(
1 - 2*x)^(7/2))/112

________________________________________________________________________________________

Rubi [A]  time = 0.013909, 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} \[ -\frac{225}{112} (1-2 x)^{7/2}+\frac{51}{2} (1-2 x)^{5/2}-\frac{3467}{24} (1-2 x)^{3/2}+\frac{1309}{2} \sqrt{1-2 x}+\frac{5929}{16 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5929/(16*Sqrt[1 - 2*x]) + (1309*Sqrt[1 - 2*x])/2 - (3467*(1 - 2*x)^(3/2))/24 + (51*(1 - 2*x)^(5/2))/2 - (225*(
1 - 2*x)^(7/2))/112

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)^{3/2}} \, dx &=\int \left (\frac{5929}{16 (1-2 x)^{3/2}}-\frac{1309}{2 \sqrt{1-2 x}}+\frac{3467}{8} \sqrt{1-2 x}-\frac{255}{2} (1-2 x)^{3/2}+\frac{225}{16} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac{5929}{16 \sqrt{1-2 x}}+\frac{1309}{2} \sqrt{1-2 x}-\frac{3467}{24} (1-2 x)^{3/2}+\frac{51}{2} (1-2 x)^{5/2}-\frac{225}{112} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0131212, size = 33, normalized size = 0.5 \[ \frac{-675 x^4-2934 x^3-6721 x^2-18230 x+18986}{21 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18986 - 18230*x - 6721*x^2 - 2934*x^3 - 675*x^4)/(21*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 30, normalized size = 0.5 \begin{align*} -{\frac{675\,{x}^{4}+2934\,{x}^{3}+6721\,{x}^{2}+18230\,x-18986}{21}{\frac{1}{\sqrt{1-2\,x}}}} \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)^(3/2),x)

[Out]

-1/21*(675*x^4+2934*x^3+6721*x^2+18230*x-18986)/(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.19176, size = 62, normalized size = 0.94 \begin{align*} -\frac{225}{112} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{51}{2} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{3467}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1309}{2} \, \sqrt{-2 \, x + 1} + \frac{5929}{16 \, \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)^(3/2),x, algorithm="maxima")

[Out]

-225/112*(-2*x + 1)^(7/2) + 51/2*(-2*x + 1)^(5/2) - 3467/24*(-2*x + 1)^(3/2) + 1309/2*sqrt(-2*x + 1) + 5929/16
/sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.52847, size = 109, normalized size = 1.65 \begin{align*} \frac{{\left (675 \, x^{4} + 2934 \, x^{3} + 6721 \, x^{2} + 18230 \, x - 18986\right )} \sqrt{-2 \, x + 1}}{21 \,{\left (2 \, 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)^(3/2),x, algorithm="fricas")

[Out]

1/21*(675*x^4 + 2934*x^3 + 6721*x^2 + 18230*x - 18986)*sqrt(-2*x + 1)/(2*x - 1)

________________________________________________________________________________________

Sympy [A]  time = 16.4108, size = 58, normalized size = 0.88 \begin{align*} - \frac{225 \left (1 - 2 x\right )^{\frac{7}{2}}}{112} + \frac{51 \left (1 - 2 x\right )^{\frac{5}{2}}}{2} - \frac{3467 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} + \frac{1309 \sqrt{1 - 2 x}}{2} + \frac{5929}{16 \sqrt{1 - 2 x}} \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)**(3/2),x)

[Out]

-225*(1 - 2*x)**(7/2)/112 + 51*(1 - 2*x)**(5/2)/2 - 3467*(1 - 2*x)**(3/2)/24 + 1309*sqrt(1 - 2*x)/2 + 5929/(16
*sqrt(1 - 2*x))

________________________________________________________________________________________

Giac [A]  time = 1.96152, size = 81, normalized size = 1.23 \begin{align*} \frac{225}{112} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{51}{2} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{3467}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1309}{2} \, \sqrt{-2 \, x + 1} + \frac{5929}{16 \, \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)^(3/2),x, algorithm="giac")

[Out]

225/112*(2*x - 1)^3*sqrt(-2*x + 1) + 51/2*(2*x - 1)^2*sqrt(-2*x + 1) - 3467/24*(-2*x + 1)^(3/2) + 1309/2*sqrt(
-2*x + 1) + 5929/16/sqrt(-2*x + 1)