3.2071 \(\int \frac{(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{1215}{704} (1-2 x)^{11/2}+\frac{117}{4} (1-2 x)^{9/2}-\frac{13905}{64} (1-2 x)^{7/2}+\frac{7497}{8} (1-2 x)^{5/2}-\frac{173215}{64} (1-2 x)^{3/2}+\frac{60025}{8} \sqrt{1-2 x}+\frac{184877}{64 \sqrt{1-2 x}} \]

[Out]

184877/(64*Sqrt[1 - 2*x]) + (60025*Sqrt[1 - 2*x])/8 - (173215*(1 - 2*x)^(3/2))/64 + (7497*(1 - 2*x)^(5/2))/8 -
(13905*(1 - 2*x)^(7/2))/64 + (117*(1 - 2*x)^(9/2))/4 - (1215*(1 - 2*x)^(11/2))/704

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Rubi [A]  time = 0.0158934, antiderivative size = 92, 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} \[ -\frac{1215}{704} (1-2 x)^{11/2}+\frac{117}{4} (1-2 x)^{9/2}-\frac{13905}{64} (1-2 x)^{7/2}+\frac{7497}{8} (1-2 x)^{5/2}-\frac{173215}{64} (1-2 x)^{3/2}+\frac{60025}{8} \sqrt{1-2 x}+\frac{184877}{64 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

184877/(64*Sqrt[1 - 2*x]) + (60025*Sqrt[1 - 2*x])/8 - (173215*(1 - 2*x)^(3/2))/64 + (7497*(1 - 2*x)^(5/2))/8 -
(13905*(1 - 2*x)^(7/2))/64 + (117*(1 - 2*x)^(9/2))/4 - (1215*(1 - 2*x)^(11/2))/704

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)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{184877}{64 (1-2 x)^{3/2}}-\frac{60025}{8 \sqrt{1-2 x}}+\frac{519645}{64} \sqrt{1-2 x}-\frac{37485}{8} (1-2 x)^{3/2}+\frac{97335}{64} (1-2 x)^{5/2}-\frac{1053}{4} (1-2 x)^{7/2}+\frac{1215}{64} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac{184877}{64 \sqrt{1-2 x}}+\frac{60025}{8} \sqrt{1-2 x}-\frac{173215}{64} (1-2 x)^{3/2}+\frac{7497}{8} (1-2 x)^{5/2}-\frac{13905}{64} (1-2 x)^{7/2}+\frac{117}{4} (1-2 x)^{9/2}-\frac{1215}{704} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0160357, size = 43, normalized size = 0.47 \[ \frac{-1215 x^6-6651 x^5-17055 x^4-28692 x^3-41012 x^2-91704 x+92760}{11 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(92760 - 91704*x - 41012*x^2 - 28692*x^3 - 17055*x^4 - 6651*x^5 - 1215*x^6)/(11*Sqrt[1 - 2*x])

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{1215\,{x}^{6}+6651\,{x}^{5}+17055\,{x}^{4}+28692\,{x}^{3}+41012\,{x}^{2}+91704\,x-92760}{11}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11*(1215*x^6+6651*x^5+17055*x^4+28692*x^3+41012*x^2+91704*x-92760)/(1-2*x)^(1/2)

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Maxima [A]  time = 2.46058, size = 86, normalized size = 0.93 \begin{align*} -\frac{1215}{704} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{117}{4} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{13905}{64} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{7497}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{173215}{64} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{60025}{8} \, \sqrt{-2 \, x + 1} + \frac{184877}{64 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215/704*(-2*x + 1)^(11/2) + 117/4*(-2*x + 1)^(9/2) - 13905/64*(-2*x + 1)^(7/2) + 7497/8*(-2*x + 1)^(5/2) - 1
73215/64*(-2*x + 1)^(3/2) + 60025/8*sqrt(-2*x + 1) + 184877/64/sqrt(-2*x + 1)

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Fricas [A]  time = 1.60454, size = 144, normalized size = 1.57 \begin{align*} \frac{{\left (1215 \, x^{6} + 6651 \, x^{5} + 17055 \, x^{4} + 28692 \, x^{3} + 41012 \, x^{2} + 91704 \, x - 92760\right )} \sqrt{-2 \, x + 1}}{11 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*(1215*x^6 + 6651*x^5 + 17055*x^4 + 28692*x^3 + 41012*x^2 + 91704*x - 92760)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 29.0217, size = 82, normalized size = 0.89 \begin{align*} - \frac{1215 \left (1 - 2 x\right )^{\frac{11}{2}}}{704} + \frac{117 \left (1 - 2 x\right )^{\frac{9}{2}}}{4} - \frac{13905 \left (1 - 2 x\right )^{\frac{7}{2}}}{64} + \frac{7497 \left (1 - 2 x\right )^{\frac{5}{2}}}{8} - \frac{173215 \left (1 - 2 x\right )^{\frac{3}{2}}}{64} + \frac{60025 \sqrt{1 - 2 x}}{8} + \frac{184877}{64 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215*(1 - 2*x)**(11/2)/704 + 117*(1 - 2*x)**(9/2)/4 - 13905*(1 - 2*x)**(7/2)/64 + 7497*(1 - 2*x)**(5/2)/8 - 1
73215*(1 - 2*x)**(3/2)/64 + 60025*sqrt(1 - 2*x)/8 + 184877/(64*sqrt(1 - 2*x))

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Giac [A]  time = 1.20078, size = 124, normalized size = 1.35 \begin{align*} \frac{1215}{704} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{117}{4} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{13905}{64} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{7497}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{173215}{64} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{60025}{8} \, \sqrt{-2 \, x + 1} + \frac{184877}{64 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1215/704*(2*x - 1)^5*sqrt(-2*x + 1) + 117/4*(2*x - 1)^4*sqrt(-2*x + 1) + 13905/64*(2*x - 1)^3*sqrt(-2*x + 1) +
7497/8*(2*x - 1)^2*sqrt(-2*x + 1) - 173215/64*(-2*x + 1)^(3/2) + 60025/8*sqrt(-2*x + 1) + 184877/64/sqrt(-2*x
+ 1)