3.2094 \(\int \frac{(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{10125 (1-2 x)^{13/2}}{1664}-\frac{161325 (1-2 x)^{11/2}}{1408}+\frac{122385}{128} (1-2 x)^{9/2}-\frac{4177401}{896} (1-2 x)^{7/2}+\frac{9504551}{640} (1-2 x)^{5/2}-\frac{4324397}{128} (1-2 x)^{3/2}+\frac{9836211}{128} \sqrt{1-2 x}+\frac{3195731}{128 \sqrt{1-2 x}} \]

[Out]

3195731/(128*Sqrt[1 - 2*x]) + (9836211*Sqrt[1 - 2*x])/128 - (4324397*(1 - 2*x)^(3/2))/128 + (9504551*(1 - 2*x)
^(5/2))/640 - (4177401*(1 - 2*x)^(7/2))/896 + (122385*(1 - 2*x)^(9/2))/128 - (161325*(1 - 2*x)^(11/2))/1408 +
(10125*(1 - 2*x)^(13/2))/1664

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Rubi [A]  time = 0.0196299, antiderivative size = 105, 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} \[ \frac{10125 (1-2 x)^{13/2}}{1664}-\frac{161325 (1-2 x)^{11/2}}{1408}+\frac{122385}{128} (1-2 x)^{9/2}-\frac{4177401}{896} (1-2 x)^{7/2}+\frac{9504551}{640} (1-2 x)^{5/2}-\frac{4324397}{128} (1-2 x)^{3/2}+\frac{9836211}{128} \sqrt{1-2 x}+\frac{3195731}{128 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3195731/(128*Sqrt[1 - 2*x]) + (9836211*Sqrt[1 - 2*x])/128 - (4324397*(1 - 2*x)^(3/2))/128 + (9504551*(1 - 2*x)
^(5/2))/640 - (4177401*(1 - 2*x)^(7/2))/896 + (122385*(1 - 2*x)^(9/2))/128 - (161325*(1 - 2*x)^(11/2))/1408 +
(10125*(1 - 2*x)^(13/2))/1664

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)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{3195731}{128 (1-2 x)^{3/2}}-\frac{9836211}{128 \sqrt{1-2 x}}+\frac{12973191}{128} \sqrt{1-2 x}-\frac{9504551}{128} (1-2 x)^{3/2}+\frac{4177401}{128} (1-2 x)^{5/2}-\frac{1101465}{128} (1-2 x)^{7/2}+\frac{161325}{128} (1-2 x)^{9/2}-\frac{10125}{128} (1-2 x)^{11/2}\right ) \, dx\\ &=\frac{3195731}{128 \sqrt{1-2 x}}+\frac{9836211}{128} \sqrt{1-2 x}-\frac{4324397}{128} (1-2 x)^{3/2}+\frac{9504551}{640} (1-2 x)^{5/2}-\frac{4177401}{896} (1-2 x)^{7/2}+\frac{122385}{128} (1-2 x)^{9/2}-\frac{161325 (1-2 x)^{11/2}}{1408}+\frac{10125 (1-2 x)^{13/2}}{1664}\\ \end{align*}

Mathematica [A]  time = 0.0194252, size = 48, normalized size = 0.46 \[ \frac{-3898125 x^7-23058000 x^6-63495075 x^5-111095730 x^4-147527176 x^3-184884496 x^2-393552752 x+395714912}{5005 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(395714912 - 393552752*x - 184884496*x^2 - 147527176*x^3 - 111095730*x^4 - 63495075*x^5 - 23058000*x^6 - 38981
25*x^7)/(5005*Sqrt[1 - 2*x])

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{3898125\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x-395714912}{5005}{\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)^3/(1-2*x)^(3/2),x)

[Out]

-1/5005*(3898125*x^7+23058000*x^6+63495075*x^5+111095730*x^4+147527176*x^3+184884496*x^2+393552752*x-395714912
)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.0333, size = 99, normalized size = 0.94 \begin{align*} \frac{10125}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{161325}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{122385}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{4177401}{896} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{9504551}{640} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{4324397}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{9836211}{128} \, \sqrt{-2 \, x + 1} + \frac{3195731}{128 \, \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)^3/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

10125/1664*(-2*x + 1)^(13/2) - 161325/1408*(-2*x + 1)^(11/2) + 122385/128*(-2*x + 1)^(9/2) - 4177401/896*(-2*x
+ 1)^(7/2) + 9504551/640*(-2*x + 1)^(5/2) - 4324397/128*(-2*x + 1)^(3/2) + 9836211/128*sqrt(-2*x + 1) + 31957
31/128/sqrt(-2*x + 1)

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Fricas [A]  time = 1.54614, size = 204, normalized size = 1.94 \begin{align*} \frac{{\left (3898125 \, x^{7} + 23058000 \, x^{6} + 63495075 \, x^{5} + 111095730 \, x^{4} + 147527176 \, x^{3} + 184884496 \, x^{2} + 393552752 \, x - 395714912\right )} \sqrt{-2 \, x + 1}}{5005 \,{\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)^3/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/5005*(3898125*x^7 + 23058000*x^6 + 63495075*x^5 + 111095730*x^4 + 147527176*x^3 + 184884496*x^2 + 393552752*
x - 395714912)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 36.3524, size = 94, normalized size = 0.9 \begin{align*} \frac{10125 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} - \frac{161325 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} + \frac{122385 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} - \frac{4177401 \left (1 - 2 x\right )^{\frac{7}{2}}}{896} + \frac{9504551 \left (1 - 2 x\right )^{\frac{5}{2}}}{640} - \frac{4324397 \left (1 - 2 x\right )^{\frac{3}{2}}}{128} + \frac{9836211 \sqrt{1 - 2 x}}{128} + \frac{3195731}{128 \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)**3/(1-2*x)**(3/2),x)

[Out]

10125*(1 - 2*x)**(13/2)/1664 - 161325*(1 - 2*x)**(11/2)/1408 + 122385*(1 - 2*x)**(9/2)/128 - 4177401*(1 - 2*x)
**(7/2)/896 + 9504551*(1 - 2*x)**(5/2)/640 - 4324397*(1 - 2*x)**(3/2)/128 + 9836211*sqrt(1 - 2*x)/128 + 319573
1/(128*sqrt(1 - 2*x))

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Giac [A]  time = 2.35574, size = 146, normalized size = 1.39 \begin{align*} \frac{10125}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{161325}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{122385}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{4177401}{896} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{9504551}{640} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{4324397}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{9836211}{128} \, \sqrt{-2 \, x + 1} + \frac{3195731}{128 \, \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)^3/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

10125/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 161325/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 122385/128*(2*x - 1)^4*sqrt(-
2*x + 1) + 4177401/896*(2*x - 1)^3*sqrt(-2*x + 1) + 9504551/640*(2*x - 1)^2*sqrt(-2*x + 1) - 4324397/128*(-2*x
+ 1)^(3/2) + 9836211/128*sqrt(-2*x + 1) + 3195731/128/sqrt(-2*x + 1)