3.20.97 \(\int \frac {(2+3 x)^4 (3+5 x)}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac {405}{224} (1-2 x)^{7/2}-\frac {4671}{160} (1-2 x)^{5/2}+\frac {3591}{16} (1-2 x)^{3/2}-\frac {24843}{16} \sqrt {1-2 x}-\frac {57281}{32 \sqrt {1-2 x}}+\frac {26411}{96 (1-2 x)^{3/2}} \]

________________________________________________________________________________________

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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} \frac {405}{224} (1-2 x)^{7/2}-\frac {4671}{160} (1-2 x)^{5/2}+\frac {3591}{16} (1-2 x)^{3/2}-\frac {24843}{16} \sqrt {1-2 x}-\frac {57281}{32 \sqrt {1-2 x}}+\frac {26411}{96 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^4*(3 + 5*x))/(1 - 2*x)^(5/2),x]

[Out]

26411/(96*(1 - 2*x)^(3/2)) - 57281/(32*Sqrt[1 - 2*x]) - (24843*Sqrt[1 - 2*x])/16 + (3591*(1 - 2*x)^(3/2))/16 -
(4671*(1 - 2*x)^(5/2))/160 + (405*(1 - 2*x)^(7/2))/224

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {26411}{32 (1-2 x)^{5/2}}-\frac {57281}{32 (1-2 x)^{3/2}}+\frac {24843}{16 \sqrt {1-2 x}}-\frac {10773}{16} \sqrt {1-2 x}+\frac {4671}{32} (1-2 x)^{3/2}-\frac {405}{32} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac {26411}{96 (1-2 x)^{3/2}}-\frac {57281}{32 \sqrt {1-2 x}}-\frac {24843}{16} \sqrt {1-2 x}+\frac {3591}{16} (1-2 x)^{3/2}-\frac {4671}{160} (1-2 x)^{5/2}+\frac {405}{224} (1-2 x)^{7/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {6075 x^5+33858 x^4+105624 x^3+435312 x^2-909264 x+301408}{105 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^4*(3 + 5*x))/(1 - 2*x)^(5/2),x]

[Out]

-1/105*(301408 - 909264*x + 435312*x^2 + 105624*x^3 + 33858*x^4 + 6075*x^5)/(1 - 2*x)^(3/2)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.03, size = 58, normalized size = 0.73 \begin {gather*} \frac {6075 (1-2 x)^5-98091 (1-2 x)^4+754110 (1-2 x)^3-5217030 (1-2 x)^2-6014505 (1-2 x)+924385}{3360 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + 3*x)^4*(3 + 5*x))/(1 - 2*x)^(5/2),x]

[Out]

(924385 - 6014505*(1 - 2*x) - 5217030*(1 - 2*x)^2 + 754110*(1 - 2*x)^3 - 98091*(1 - 2*x)^4 + 6075*(1 - 2*x)^5)
/(3360*(1 - 2*x)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 1.15, size = 46, normalized size = 0.58 \begin {gather*} -\frac {{\left (6075 \, x^{5} + 33858 \, x^{4} + 105624 \, x^{3} + 435312 \, x^{2} - 909264 \, x + 301408\right )} \sqrt {-2 \, x + 1}}{105 \, {\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)^4*(3+5*x)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/105*(6075*x^5 + 33858*x^4 + 105624*x^3 + 435312*x^2 - 909264*x + 301408)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

________________________________________________________________________________________

giac [A]  time = 1.21, size = 72, normalized size = 0.91 \begin {gather*} -\frac {405}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {4671}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {3591}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {24843}{16} \, \sqrt {-2 \, x + 1} - \frac {343 \, {\left (501 \, x - 212\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)^4*(3+5*x)/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-405/224*(2*x - 1)^3*sqrt(-2*x + 1) - 4671/160*(2*x - 1)^2*sqrt(-2*x + 1) + 3591/16*(-2*x + 1)^(3/2) - 24843/1
6*sqrt(-2*x + 1) - 343/48*(501*x - 212)/((2*x - 1)*sqrt(-2*x + 1))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {6075 x^{5}+33858 x^{4}+105624 x^{3}+435312 x^{2}-909264 x +301408}{105 \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)^4*(5*x+3)/(-2*x+1)^(5/2),x)

[Out]

-1/105*(6075*x^5+33858*x^4+105624*x^3+435312*x^2-909264*x+301408)/(-2*x+1)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 51, normalized size = 0.65 \begin {gather*} \frac {405}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {4671}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {3591}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {24843}{16} \, \sqrt {-2 \, x + 1} + \frac {343 \, {\left (501 \, x - 212\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)^4*(3+5*x)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

405/224*(-2*x + 1)^(7/2) - 4671/160*(-2*x + 1)^(5/2) + 3591/16*(-2*x + 1)^(3/2) - 24843/16*sqrt(-2*x + 1) + 34
3/48*(501*x - 212)/(-2*x + 1)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 50, normalized size = 0.63 \begin {gather*} \frac {\frac {57281\,x}{16}-\frac {18179}{12}}{{\left (1-2\,x\right )}^{3/2}}-\frac {24843\,\sqrt {1-2\,x}}{16}+\frac {3591\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {4671\,{\left (1-2\,x\right )}^{5/2}}{160}+\frac {405\,{\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)^4*(5*x + 3))/(1 - 2*x)^(5/2),x)

[Out]

((57281*x)/16 - 18179/12)/(1 - 2*x)^(3/2) - (24843*(1 - 2*x)^(1/2))/16 + (3591*(1 - 2*x)^(3/2))/16 - (4671*(1
- 2*x)^(5/2))/160 + (405*(1 - 2*x)^(7/2))/224

________________________________________________________________________________________

sympy [A]  time = 30.35, size = 70, normalized size = 0.89 \begin {gather*} \frac {405 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} - \frac {4671 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} + \frac {3591 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} - \frac {24843 \sqrt {1 - 2 x}}{16} - \frac {57281}{32 \sqrt {1 - 2 x}} + \frac {26411}{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)**4*(3+5*x)/(1-2*x)**(5/2),x)

[Out]

405*(1 - 2*x)**(7/2)/224 - 4671*(1 - 2*x)**(5/2)/160 + 3591*(1 - 2*x)**(3/2)/16 - 24843*sqrt(1 - 2*x)/16 - 572
81/(32*sqrt(1 - 2*x)) + 26411/(96*(1 - 2*x)**(3/2))

________________________________________________________________________________________