3.2158 \(\int \frac{(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{10125 (1-2 x)^{11/2}}{1408}-\frac{17925}{128} (1-2 x)^{9/2}+\frac{1101465}{896} (1-2 x)^{7/2}-\frac{4177401}{640} (1-2 x)^{5/2}+\frac{9504551}{384} (1-2 x)^{3/2}-\frac{12973191}{128} \sqrt{1-2 x}-\frac{9836211}{128 \sqrt{1-2 x}}+\frac{3195731}{384 (1-2 x)^{3/2}} \]

[Out]

3195731/(384*(1 - 2*x)^(3/2)) - 9836211/(128*Sqrt[1 - 2*x]) - (12973191*Sqrt[1 - 2*x])/128 + (9504551*(1 - 2*x
)^(3/2))/384 - (4177401*(1 - 2*x)^(5/2))/640 + (1101465*(1 - 2*x)^(7/2))/896 - (17925*(1 - 2*x)^(9/2))/128 + (
10125*(1 - 2*x)^(11/2))/1408

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Rubi [A]  time = 0.0191662, 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} \[ \frac{10125 (1-2 x)^{11/2}}{1408}-\frac{17925}{128} (1-2 x)^{9/2}+\frac{1101465}{896} (1-2 x)^{7/2}-\frac{4177401}{640} (1-2 x)^{5/2}+\frac{9504551}{384} (1-2 x)^{3/2}-\frac{12973191}{128} \sqrt{1-2 x}-\frac{9836211}{128 \sqrt{1-2 x}}+\frac{3195731}{384 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3195731/(384*(1 - 2*x)^(3/2)) - 9836211/(128*Sqrt[1 - 2*x]) - (12973191*Sqrt[1 - 2*x])/128 + (9504551*(1 - 2*x
)^(3/2))/384 - (4177401*(1 - 2*x)^(5/2))/640 + (1101465*(1 - 2*x)^(7/2))/896 - (17925*(1 - 2*x)^(9/2))/128 + (
10125*(1 - 2*x)^(11/2))/1408

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)^{5/2}} \, dx &=\int \left (\frac{3195731}{128 (1-2 x)^{5/2}}-\frac{9836211}{128 (1-2 x)^{3/2}}+\frac{12973191}{128 \sqrt{1-2 x}}-\frac{9504551}{128} \sqrt{1-2 x}+\frac{4177401}{128} (1-2 x)^{3/2}-\frac{1101465}{128} (1-2 x)^{5/2}+\frac{161325}{128} (1-2 x)^{7/2}-\frac{10125}{128} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac{3195731}{384 (1-2 x)^{3/2}}-\frac{9836211}{128 \sqrt{1-2 x}}-\frac{12973191}{128} \sqrt{1-2 x}+\frac{9504551}{384} (1-2 x)^{3/2}-\frac{4177401}{640} (1-2 x)^{5/2}+\frac{1101465}{896} (1-2 x)^{7/2}-\frac{17925}{128} (1-2 x)^{9/2}+\frac{10125 (1-2 x)^{11/2}}{1408}\\ \end{align*}

Mathematica [A]  time = 0.0206236, size = 48, normalized size = 0.46 \[ -\frac{1063125 x^7+6630750 x^6+19961775 x^5+41201532 x^4+77493296 x^3+258342648 x^2-522173856 x+173891632}{1155 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(173891632 - 522173856*x + 258342648*x^2 + 77493296*x^3 + 41201532*x^4 + 19961775*x^5 + 6630750*x^6 + 1063125
*x^7)/(1155*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{1063125\,{x}^{7}+6630750\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-522173856\,x+173891632}{1155} \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)^4*(3+5*x)^3/(1-2*x)^(5/2),x)

[Out]

-1/1155*(1063125*x^7+6630750*x^6+19961775*x^5+41201532*x^4+77493296*x^3+258342648*x^2-522173856*x+173891632)/(
1-2*x)^(3/2)

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Maxima [A]  time = 2.13053, size = 93, normalized size = 0.89 \begin{align*} \frac{10125}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{17925}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{1101465}{896} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{4177401}{640} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{9504551}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{12973191}{128} \, \sqrt{-2 \, x + 1} + \frac{41503 \,{\left (711 \, x - 317\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)^4*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

10125/1408*(-2*x + 1)^(11/2) - 17925/128*(-2*x + 1)^(9/2) + 1101465/896*(-2*x + 1)^(7/2) - 4177401/640*(-2*x +
1)^(5/2) + 9504551/384*(-2*x + 1)^(3/2) - 12973191/128*sqrt(-2*x + 1) + 41503/192*(711*x - 317)/(-2*x + 1)^(3
/2)

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Fricas [A]  time = 1.58137, size = 212, normalized size = 2.02 \begin{align*} -\frac{{\left (1063125 \, x^{7} + 6630750 \, x^{6} + 19961775 \, x^{5} + 41201532 \, x^{4} + 77493296 \, x^{3} + 258342648 \, x^{2} - 522173856 \, x + 173891632\right )} \sqrt{-2 \, x + 1}}{1155 \,{\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)^4*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/1155*(1063125*x^7 + 6630750*x^6 + 19961775*x^5 + 41201532*x^4 + 77493296*x^3 + 258342648*x^2 - 522173856*x
+ 173891632)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 30.3876, size = 94, normalized size = 0.9 \begin{align*} \frac{10125 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} - \frac{17925 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} + \frac{1101465 \left (1 - 2 x\right )^{\frac{7}{2}}}{896} - \frac{4177401 \left (1 - 2 x\right )^{\frac{5}{2}}}{640} + \frac{9504551 \left (1 - 2 x\right )^{\frac{3}{2}}}{384} - \frac{12973191 \sqrt{1 - 2 x}}{128} - \frac{9836211}{128 \sqrt{1 - 2 x}} + \frac{3195731}{384 \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)**4*(3+5*x)**3/(1-2*x)**(5/2),x)

[Out]

10125*(1 - 2*x)**(11/2)/1408 - 17925*(1 - 2*x)**(9/2)/128 + 1101465*(1 - 2*x)**(7/2)/896 - 4177401*(1 - 2*x)**
(5/2)/640 + 9504551*(1 - 2*x)**(3/2)/384 - 12973191*sqrt(1 - 2*x)/128 - 9836211/(128*sqrt(1 - 2*x)) + 3195731/
(384*(1 - 2*x)**(3/2))

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Giac [A]  time = 2.47305, size = 140, normalized size = 1.33 \begin{align*} -\frac{10125}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{17925}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{1101465}{896} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{4177401}{640} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{9504551}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{12973191}{128} \, \sqrt{-2 \, x + 1} - \frac{41503 \,{\left (711 \, x - 317\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)^4*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-10125/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 17925/128*(2*x - 1)^4*sqrt(-2*x + 1) - 1101465/896*(2*x - 1)^3*sqrt(-
2*x + 1) - 4177401/640*(2*x - 1)^2*sqrt(-2*x + 1) + 9504551/384*(-2*x + 1)^(3/2) - 12973191/128*sqrt(-2*x + 1)
- 41503/192*(711*x - 317)/((2*x - 1)*sqrt(-2*x + 1))