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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, 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 {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}} \end {gather*}

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 = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {3375 x^5+18180 x^4+55145 x^3+223458 x^2-465060 x+154264}{21 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 58, normalized size = 0.73 \begin {gather*} \frac {3375 (1-2 x)^5-53235 (1-2 x)^4+399770 (1-2 x)^3-2701314 (1-2 x)^2-3041577 (1-2 x)+456533}{672 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(456533 - 3041577*(1 - 2*x) - 2701314*(1 - 2*x)^2 + 399770*(1 - 2*x)^3 - 53235*(1 - 2*x)^4 + 3375*(1 - 2*x)^5)
/(672*(1 - 2*x)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 1.20, size = 46, normalized size = 0.58 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 1.26, size = 72, normalized size = 0.91 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {3375 x^{5}+18180 x^{4}+55145 x^{3}+223458 x^{2}-465060 x +154264}{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)^2*(5*x+3)^3/(-2*x+1)^(5/2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.62, size = 51, normalized size = 0.65 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 50, normalized size = 0.63 \begin {gather*} \frac {\frac {144837\,x}{16}-\frac {92323}{24}}{{\left (1-2\,x\right )}^{3/2}}-\frac {64317\,\sqrt {1-2\,x}}{16}+\frac {28555\,{\left (1-2\,x\right )}^{3/2}}{48}-\frac {2535\,{\left (1-2\,x\right )}^{5/2}}{32}+\frac {1125\,{\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)^2*(5*x + 3)^3)/(1 - 2*x)^(5/2),x)

[Out]

((144837*x)/16 - 92323/24)/(1 - 2*x)^(3/2) - (64317*(1 - 2*x)^(1/2))/16 + (28555*(1 - 2*x)^(3/2))/48 - (2535*(
1 - 2*x)^(5/2))/32 + (1125*(1 - 2*x)^(7/2))/224

________________________________________________________________________________________

sympy [A]  time = 28.93, size = 70, normalized size = 0.89 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________