3.2084 \(\int \frac{(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2025}{704} (1-2 x)^{11/2}+\frac{1545}{32} (1-2 x)^{9/2}-\frac{159111}{448} (1-2 x)^{7/2}+\frac{121359}{80} (1-2 x)^{5/2}-\frac{832951}{192} (1-2 x)^{3/2}+\frac{381073}{32} \sqrt{1-2 x}+\frac{290521}{64 \sqrt{1-2 x}} \]

[Out]

290521/(64*Sqrt[1 - 2*x]) + (381073*Sqrt[1 - 2*x])/32 - (832951*(1 - 2*x)^(3/2))/192 + (121359*(1 - 2*x)^(5/2)
)/80 - (159111*(1 - 2*x)^(7/2))/448 + (1545*(1 - 2*x)^(9/2))/32 - (2025*(1 - 2*x)^(11/2))/704

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Rubi [A]  time = 0.0182143, antiderivative size = 92, normalized size of antiderivative = 1., 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} \[ -\frac{2025}{704} (1-2 x)^{11/2}+\frac{1545}{32} (1-2 x)^{9/2}-\frac{159111}{448} (1-2 x)^{7/2}+\frac{121359}{80} (1-2 x)^{5/2}-\frac{832951}{192} (1-2 x)^{3/2}+\frac{381073}{32} \sqrt{1-2 x}+\frac{290521}{64 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

290521/(64*Sqrt[1 - 2*x]) + (381073*Sqrt[1 - 2*x])/32 - (832951*(1 - 2*x)^(3/2))/192 + (121359*(1 - 2*x)^(5/2)
)/80 - (159111*(1 - 2*x)^(7/2))/448 + (1545*(1 - 2*x)^(9/2))/32 - (2025*(1 - 2*x)^(11/2))/704

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)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{290521}{64 (1-2 x)^{3/2}}-\frac{381073}{32 \sqrt{1-2 x}}+\frac{832951}{64} \sqrt{1-2 x}-\frac{121359}{16} (1-2 x)^{3/2}+\frac{159111}{64} (1-2 x)^{5/2}-\frac{13905}{32} (1-2 x)^{7/2}+\frac{2025}{64} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac{290521}{64 \sqrt{1-2 x}}+\frac{381073}{32} \sqrt{1-2 x}-\frac{832951}{192} (1-2 x)^{3/2}+\frac{121359}{80} (1-2 x)^{5/2}-\frac{159111}{448} (1-2 x)^{7/2}+\frac{1545}{32} (1-2 x)^{9/2}-\frac{2025}{704} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.017279, size = 43, normalized size = 0.47 \[ \frac{-212625 x^6-1146600 x^5-2899485 x^4-4819932 x^3-6831172 x^2-15214664 x+15380984}{1155 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15380984 - 15214664*x - 6831172*x^2 - 4819932*x^3 - 2899485*x^4 - 1146600*x^5 - 212625*x^6)/(1155*Sqrt[1 - 2*
x])

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{212625\,{x}^{6}+1146600\,{x}^{5}+2899485\,{x}^{4}+4819932\,{x}^{3}+6831172\,{x}^{2}[phone]\,x-15380984}{1155}{\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)^2/(1-2*x)^(3/2),x)

[Out]

-1/1155*(212625*x^6+1146600*x^5+2899485*x^4+4819932*x^3+6831172*x^2+15214664*x-15380984)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.615, size = 86, normalized size = 0.93 \begin{align*} -\frac{2025}{704} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{1545}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{159111}{448} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{121359}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{832951}{192} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{381073}{32} \, \sqrt{-2 \, x + 1} + \frac{290521}{64 \, \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)^2/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

-2025/704*(-2*x + 1)^(11/2) + 1545/32*(-2*x + 1)^(9/2) - 159111/448*(-2*x + 1)^(7/2) + 121359/80*(-2*x + 1)^(5
/2) - 832951/192*(-2*x + 1)^(3/2) + 381073/32*sqrt(-2*x + 1) + 290521/64/sqrt(-2*x + 1)

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Fricas [A]  time = 1.68885, size = 170, normalized size = 1.85 \begin{align*} \frac{{\left (212625 \, x^{6} + 1146600 \, x^{5} + 2899485 \, x^{4} + 4819932 \, x^{3} + 6831172 \, x^{2} + 15214664 \, x - 15380984\right )} \sqrt{-2 \, x + 1}}{1155 \,{\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)^2/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/1155*(212625*x^6 + 1146600*x^5 + 2899485*x^4 + 4819932*x^3 + 6831172*x^2 + 15214664*x - 15380984)*sqrt(-2*x
+ 1)/(2*x - 1)

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Sympy [A]  time = 28.2559, size = 82, normalized size = 0.89 \begin{align*} - \frac{2025 \left (1 - 2 x\right )^{\frac{11}{2}}}{704} + \frac{1545 \left (1 - 2 x\right )^{\frac{9}{2}}}{32} - \frac{159111 \left (1 - 2 x\right )^{\frac{7}{2}}}{448} + \frac{121359 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} - \frac{832951 \left (1 - 2 x\right )^{\frac{3}{2}}}{192} + \frac{381073 \sqrt{1 - 2 x}}{32} + \frac{290521}{64 \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)**2/(1-2*x)**(3/2),x)

[Out]

-2025*(1 - 2*x)**(11/2)/704 + 1545*(1 - 2*x)**(9/2)/32 - 159111*(1 - 2*x)**(7/2)/448 + 121359*(1 - 2*x)**(5/2)
/80 - 832951*(1 - 2*x)**(3/2)/192 + 381073*sqrt(1 - 2*x)/32 + 290521/(64*sqrt(1 - 2*x))

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Giac [A]  time = 1.73911, size = 124, normalized size = 1.35 \begin{align*} \frac{2025}{704} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{1545}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{159111}{448} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{121359}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{832951}{192} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{381073}{32} \, \sqrt{-2 \, x + 1} + \frac{290521}{64 \, \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)^2/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

2025/704*(2*x - 1)^5*sqrt(-2*x + 1) + 1545/32*(2*x - 1)^4*sqrt(-2*x + 1) + 159111/448*(2*x - 1)^3*sqrt(-2*x +
1) + 121359/80*(2*x - 1)^2*sqrt(-2*x + 1) - 832951/192*(-2*x + 1)^(3/2) + 381073/32*sqrt(-2*x + 1) + 290521/64
/sqrt(-2*x + 1)