3.2085 \(\int \frac{(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{75}{32} (1-2 x)^{9/2}-\frac{7695}{224} (1-2 x)^{7/2}+\frac{17541}{80} (1-2 x)^{5/2}-\frac{39977}{48} (1-2 x)^{3/2}+\frac{91091}{32} \sqrt{1-2 x}+\frac{41503}{32 \sqrt{1-2 x}} \]

[Out]

41503/(32*Sqrt[1 - 2*x]) + (91091*Sqrt[1 - 2*x])/32 - (39977*(1 - 2*x)^(3/2))/48 + (17541*(1 - 2*x)^(5/2))/80
- (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2))/32

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Rubi [A]  time = 0.0153282, antiderivative size = 79, 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} \[ \frac{75}{32} (1-2 x)^{9/2}-\frac{7695}{224} (1-2 x)^{7/2}+\frac{17541}{80} (1-2 x)^{5/2}-\frac{39977}{48} (1-2 x)^{3/2}+\frac{91091}{32} \sqrt{1-2 x}+\frac{41503}{32 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

41503/(32*Sqrt[1 - 2*x]) + (91091*Sqrt[1 - 2*x])/32 - (39977*(1 - 2*x)^(3/2))/48 + (17541*(1 - 2*x)^(5/2))/80
- (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2))/32

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)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{41503}{32 (1-2 x)^{3/2}}-\frac{91091}{32 \sqrt{1-2 x}}+\frac{39977}{16} \sqrt{1-2 x}-\frac{17541}{16} (1-2 x)^{3/2}+\frac{7695}{32} (1-2 x)^{5/2}-\frac{675}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac{41503}{32 \sqrt{1-2 x}}+\frac{91091}{32} \sqrt{1-2 x}-\frac{39977}{48} (1-2 x)^{3/2}+\frac{17541}{80} (1-2 x)^{5/2}-\frac{7695}{224} (1-2 x)^{7/2}+\frac{75}{32} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0160771, size = 38, normalized size = 0.48 \[ \frac{-7875 x^5-38025 x^4-88443 x^3-150253 x^2-359726 x+367286}{105 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(367286 - 359726*x - 150253*x^2 - 88443*x^3 - 38025*x^4 - 7875*x^5)/(105*Sqrt[1 - 2*x])

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Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{7875\,{x}^{5}+38025\,{x}^{4}+88443\,{x}^{3}+150253\,{x}^{2}+359726\,x-367286}{105}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(7875*x^5+38025*x^4+88443*x^3+150253*x^2+359726*x-367286)/(1-2*x)^(1/2)

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Maxima [A]  time = 2.99201, size = 74, normalized size = 0.94 \begin{align*} \frac{75}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{7695}{224} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{17541}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{39977}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{91091}{32} \, \sqrt{-2 \, x + 1} + \frac{41503}{32 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/32*(-2*x + 1)^(9/2) - 7695/224*(-2*x + 1)^(7/2) + 17541/80*(-2*x + 1)^(5/2) - 39977/48*(-2*x + 1)^(3/2) + 9
1091/32*sqrt(-2*x + 1) + 41503/32/sqrt(-2*x + 1)

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Fricas [A]  time = 1.50467, size = 135, normalized size = 1.71 \begin{align*} \frac{{\left (7875 \, x^{5} + 38025 \, x^{4} + 88443 \, x^{3} + 150253 \, x^{2} + 359726 \, x - 367286\right )} \sqrt{-2 \, x + 1}}{105 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(7875*x^5 + 38025*x^4 + 88443*x^3 + 150253*x^2 + 359726*x - 367286)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 21.8675, size = 70, normalized size = 0.89 \begin{align*} \frac{75 \left (1 - 2 x\right )^{\frac{9}{2}}}{32} - \frac{7695 \left (1 - 2 x\right )^{\frac{7}{2}}}{224} + \frac{17541 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} - \frac{39977 \left (1 - 2 x\right )^{\frac{3}{2}}}{48} + \frac{91091 \sqrt{1 - 2 x}}{32} + \frac{41503}{32 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75*(1 - 2*x)**(9/2)/32 - 7695*(1 - 2*x)**(7/2)/224 + 17541*(1 - 2*x)**(5/2)/80 - 39977*(1 - 2*x)**(3/2)/48 + 9
1091*sqrt(1 - 2*x)/32 + 41503/(32*sqrt(1 - 2*x))

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Giac [A]  time = 2.02404, size = 103, normalized size = 1.3 \begin{align*} \frac{75}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{7695}{224} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{17541}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{39977}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{91091}{32} \, \sqrt{-2 \, x + 1} + \frac{41503}{32 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/32*(2*x - 1)^4*sqrt(-2*x + 1) + 7695/224*(2*x - 1)^3*sqrt(-2*x + 1) + 17541/80*(2*x - 1)^2*sqrt(-2*x + 1) -
39977/48*(-2*x + 1)^(3/2) + 91091/32*sqrt(-2*x + 1) + 41503/32/sqrt(-2*x + 1)