3.2072 \(\int \frac{(2+3 x)^4 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{45}{32} (1-2 x)^{9/2}-\frac{4671}{224} (1-2 x)^{7/2}+\frac{10773}{80} (1-2 x)^{5/2}-\frac{8281}{16} (1-2 x)^{3/2}+\frac{57281}{32} \sqrt{1-2 x}+\frac{26411}{32 \sqrt{1-2 x}} \]

[Out]

26411/(32*Sqrt[1 - 2*x]) + (57281*Sqrt[1 - 2*x])/32 - (8281*(1 - 2*x)^(3/2))/16 + (10773*(1 - 2*x)^(5/2))/80 -
(4671*(1 - 2*x)^(7/2))/224 + (45*(1 - 2*x)^(9/2))/32

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Rubi [A]  time = 0.0136894, antiderivative size = 79, normalized size of antiderivative = 1., 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} \[ \frac{45}{32} (1-2 x)^{9/2}-\frac{4671}{224} (1-2 x)^{7/2}+\frac{10773}{80} (1-2 x)^{5/2}-\frac{8281}{16} (1-2 x)^{3/2}+\frac{57281}{32} \sqrt{1-2 x}+\frac{26411}{32 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

26411/(32*Sqrt[1 - 2*x]) + (57281*Sqrt[1 - 2*x])/32 - (8281*(1 - 2*x)^(3/2))/16 + (10773*(1 - 2*x)^(5/2))/80 -
(4671*(1 - 2*x)^(7/2))/224 + (45*(1 - 2*x)^(9/2))/32

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)^{3/2}} \, dx &=\int \left (\frac{26411}{32 (1-2 x)^{3/2}}-\frac{57281}{32 \sqrt{1-2 x}}+\frac{24843}{16} \sqrt{1-2 x}-\frac{10773}{16} (1-2 x)^{3/2}+\frac{4671}{32} (1-2 x)^{5/2}-\frac{405}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac{26411}{32 \sqrt{1-2 x}}+\frac{57281}{32} \sqrt{1-2 x}-\frac{8281}{16} (1-2 x)^{3/2}+\frac{10773}{80} (1-2 x)^{5/2}-\frac{4671}{224} (1-2 x)^{7/2}+\frac{45}{32} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0143781, size = 38, normalized size = 0.48 \[ \frac{-1575 x^5-7740 x^4-18288 x^3-31448 x^2-75776 x+77456}{35 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(77456 - 75776*x - 31448*x^2 - 18288*x^3 - 7740*x^4 - 1575*x^5)/(35*Sqrt[1 - 2*x])

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Maple [A]  time = 0.002, size = 35, normalized size = 0.4 \begin{align*} -{\frac{1575\,{x}^{5}+7740\,{x}^{4}+18288\,{x}^{3}+31448\,{x}^{2}+75776\,x-77456}{35}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4*(3+5*x)/(1-2*x)^(3/2),x)

[Out]

-1/35*(1575*x^5+7740*x^4+18288*x^3+31448*x^2+75776*x-77456)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.15644, size = 74, normalized size = 0.94 \begin{align*} \frac{45}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{4671}{224} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{10773}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{8281}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{57281}{32} \, \sqrt{-2 \, x + 1} + \frac{26411}{32 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

45/32*(-2*x + 1)^(9/2) - 4671/224*(-2*x + 1)^(7/2) + 10773/80*(-2*x + 1)^(5/2) - 8281/16*(-2*x + 1)^(3/2) + 57
281/32*sqrt(-2*x + 1) + 26411/32/sqrt(-2*x + 1)

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Fricas [A]  time = 1.55445, size = 128, normalized size = 1.62 \begin{align*} \frac{{\left (1575 \, x^{5} + 7740 \, x^{4} + 18288 \, x^{3} + 31448 \, x^{2} + 75776 \, x - 77456\right )} \sqrt{-2 \, x + 1}}{35 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/35*(1575*x^5 + 7740*x^4 + 18288*x^3 + 31448*x^2 + 75776*x - 77456)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 22.1817, size = 70, normalized size = 0.89 \begin{align*} \frac{45 \left (1 - 2 x\right )^{\frac{9}{2}}}{32} - \frac{4671 \left (1 - 2 x\right )^{\frac{7}{2}}}{224} + \frac{10773 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} - \frac{8281 \left (1 - 2 x\right )^{\frac{3}{2}}}{16} + \frac{57281 \sqrt{1 - 2 x}}{32} + \frac{26411}{32 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4*(3+5*x)/(1-2*x)**(3/2),x)

[Out]

45*(1 - 2*x)**(9/2)/32 - 4671*(1 - 2*x)**(7/2)/224 + 10773*(1 - 2*x)**(5/2)/80 - 8281*(1 - 2*x)**(3/2)/16 + 57
281*sqrt(1 - 2*x)/32 + 26411/(32*sqrt(1 - 2*x))

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Giac [A]  time = 2.34329, size = 103, normalized size = 1.3 \begin{align*} \frac{45}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{4671}{224} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{10773}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{8281}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{57281}{32} \, \sqrt{-2 \, x + 1} + \frac{26411}{32 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

45/32*(2*x - 1)^4*sqrt(-2*x + 1) + 4671/224*(2*x - 1)^3*sqrt(-2*x + 1) + 10773/80*(2*x - 1)^2*sqrt(-2*x + 1) -
8281/16*(-2*x + 1)^(3/2) + 57281/32*sqrt(-2*x + 1) + 26411/32/sqrt(-2*x + 1)