3.2135 \(\int \frac{(2+3 x)^5 (3+5 x)}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{135}{64} (1-2 x)^{9/2}+\frac{1053}{28} (1-2 x)^{7/2}-\frac{19467}{64} (1-2 x)^{5/2}+\frac{12495}{8} (1-2 x)^{3/2}-\frac{519645}{64} \sqrt{1-2 x}-\frac{60025}{8 \sqrt{1-2 x}}+\frac{184877}{192 (1-2 x)^{3/2}} \]

[Out]

184877/(192*(1 - 2*x)^(3/2)) - 60025/(8*Sqrt[1 - 2*x]) - (519645*Sqrt[1 - 2*x])/64 + (12495*(1 - 2*x)^(3/2))/8
- (19467*(1 - 2*x)^(5/2))/64 + (1053*(1 - 2*x)^(7/2))/28 - (135*(1 - 2*x)^(9/2))/64

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Rubi [A]  time = 0.0174089, 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} \[ -\frac{135}{64} (1-2 x)^{9/2}+\frac{1053}{28} (1-2 x)^{7/2}-\frac{19467}{64} (1-2 x)^{5/2}+\frac{12495}{8} (1-2 x)^{3/2}-\frac{519645}{64} \sqrt{1-2 x}-\frac{60025}{8 \sqrt{1-2 x}}+\frac{184877}{192 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

184877/(192*(1 - 2*x)^(3/2)) - 60025/(8*Sqrt[1 - 2*x]) - (519645*Sqrt[1 - 2*x])/64 + (12495*(1 - 2*x)^(3/2))/8
- (19467*(1 - 2*x)^(5/2))/64 + (1053*(1 - 2*x)^(7/2))/28 - (135*(1 - 2*x)^(9/2))/64

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)^{5/2}} \, dx &=\int \left (\frac{184877}{64 (1-2 x)^{5/2}}-\frac{60025}{8 (1-2 x)^{3/2}}+\frac{519645}{64 \sqrt{1-2 x}}-\frac{37485}{8} \sqrt{1-2 x}+\frac{97335}{64} (1-2 x)^{3/2}-\frac{1053}{4} (1-2 x)^{5/2}+\frac{1215}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac{184877}{192 (1-2 x)^{3/2}}-\frac{60025}{8 \sqrt{1-2 x}}-\frac{519645}{64} \sqrt{1-2 x}+\frac{12495}{8} (1-2 x)^{3/2}-\frac{19467}{64} (1-2 x)^{5/2}+\frac{1053}{28} (1-2 x)^{7/2}-\frac{135}{64} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0199507, size = 43, normalized size = 0.47 \[ -\frac{2835 x^6+16767 x^5+49653 x^4+114084 x^3+412812 x^2-844104 x+280696}{21 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(280696 - 844104*x + 412812*x^2 + 114084*x^3 + 49653*x^4 + 16767*x^5 + 2835*x^6)/(21*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.005, size = 40, normalized size = 0.4 \begin{align*} -{\frac{2835\,{x}^{6}+16767\,{x}^{5}+49653\,{x}^{4}+114084\,{x}^{3}+412812\,{x}^{2}-844104\,x+280696}{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)^5*(3+5*x)/(1-2*x)^(5/2),x)

[Out]

-1/21*(2835*x^6+16767*x^5+49653*x^4+114084*x^3+412812*x^2-844104*x+280696)/(1-2*x)^(3/2)

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Maxima [A]  time = 2.35095, size = 81, normalized size = 0.88 \begin{align*} -\frac{135}{64} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{1053}{28} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{19467}{64} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{12495}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{519645}{64} \, \sqrt{-2 \, x + 1} + \frac{2401 \,{\left (1200 \, x - 523\right )}}{192 \,{\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)^5*(3+5*x)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-135/64*(-2*x + 1)^(9/2) + 1053/28*(-2*x + 1)^(7/2) - 19467/64*(-2*x + 1)^(5/2) + 12495/8*(-2*x + 1)^(3/2) - 5
19645/64*sqrt(-2*x + 1) + 2401/192*(1200*x - 523)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.61589, size = 163, normalized size = 1.77 \begin{align*} -\frac{{\left (2835 \, x^{6} + 16767 \, x^{5} + 49653 \, x^{4} + 114084 \, x^{3} + 412812 \, x^{2} - 844104 \, x + 280696\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)^5*(3+5*x)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/21*(2835*x^6 + 16767*x^5 + 49653*x^4 + 114084*x^3 + 412812*x^2 - 844104*x + 280696)*sqrt(-2*x + 1)/(4*x^2 -
4*x + 1)

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Sympy [A]  time = 24.2465, size = 82, normalized size = 0.89 \begin{align*} - \frac{135 \left (1 - 2 x\right )^{\frac{9}{2}}}{64} + \frac{1053 \left (1 - 2 x\right )^{\frac{7}{2}}}{28} - \frac{19467 \left (1 - 2 x\right )^{\frac{5}{2}}}{64} + \frac{12495 \left (1 - 2 x\right )^{\frac{3}{2}}}{8} - \frac{519645 \sqrt{1 - 2 x}}{64} - \frac{60025}{8 \sqrt{1 - 2 x}} + \frac{184877}{192 \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)**5*(3+5*x)/(1-2*x)**(5/2),x)

[Out]

-135*(1 - 2*x)**(9/2)/64 + 1053*(1 - 2*x)**(7/2)/28 - 19467*(1 - 2*x)**(5/2)/64 + 12495*(1 - 2*x)**(3/2)/8 - 5
19645*sqrt(1 - 2*x)/64 - 60025/(8*sqrt(1 - 2*x)) + 184877/(192*(1 - 2*x)**(3/2))

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Giac [A]  time = 1.95427, size = 119, normalized size = 1.29 \begin{align*} -\frac{135}{64} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{1053}{28} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{19467}{64} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{12495}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{519645}{64} \, \sqrt{-2 \, x + 1} - \frac{2401 \,{\left (1200 \, x - 523\right )}}{192 \,{\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)^5*(3+5*x)/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-135/64*(2*x - 1)^4*sqrt(-2*x + 1) - 1053/28*(2*x - 1)^3*sqrt(-2*x + 1) - 19467/64*(2*x - 1)^2*sqrt(-2*x + 1)
+ 12495/8*(-2*x + 1)^(3/2) - 519645/64*sqrt(-2*x + 1) - 2401/192*(1200*x - 523)/((2*x - 1)*sqrt(-2*x + 1))