3.2148 \(\int \frac{(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0153137, 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} \[ \frac{675}{224} (1-2 x)^{7/2}-\frac{1539}{32} (1-2 x)^{5/2}+\frac{5847}{16} (1-2 x)^{3/2}-\frac{39977}{16} \sqrt{1-2 x}-\frac{91091}{32 \sqrt{1-2 x}}+\frac{41503}{96 (1-2 x)^{3/2}} \]

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*}

Mathematica [A]  time = 0.0147079, size = 38, normalized size = 0.48 \[ -\frac{2025 x^5+11097 x^4+34137 x^3+139497 x^2-290838 x+96442}{21 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{2025\,{x}^{5}+11097\,{x}^{4}+34137\,{x}^{3}+139497\,{x}^{2}-290838\,x+96442}{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)^3*(3+5*x)^2/(1-2*x)^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0491, size = 69, normalized size = 0.87 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.58872, size = 144, normalized size = 1.82 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 19.8033, size = 70, normalized size = 0.89 \begin{align*} \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{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 2.52176, size = 97, normalized size = 1.23 \begin{align*} -\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{align*}

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))