3.2013 \(\int \frac{(2+3 x)^4 (3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2025}{832} (1-2 x)^{13/2}+\frac{13905}{352} (1-2 x)^{11/2}-\frac{17679}{64} (1-2 x)^{9/2}+\frac{17337}{16} (1-2 x)^{7/2}-\frac{832951}{320} (1-2 x)^{5/2}+\frac{381073}{96} (1-2 x)^{3/2}-\frac{290521}{64} \sqrt{1-2 x} \]

[Out]

(-290521*Sqrt[1 - 2*x])/64 + (381073*(1 - 2*x)^(3/2))/96 - (832951*(1 - 2*x)^(5/2))/320 + (17337*(1 - 2*x)^(7/
2))/16 - (17679*(1 - 2*x)^(9/2))/64 + (13905*(1 - 2*x)^(11/2))/352 - (2025*(1 - 2*x)^(13/2))/832

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Rubi [A]  time = 0.0171868, 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} \[ -\frac{2025}{832} (1-2 x)^{13/2}+\frac{13905}{352} (1-2 x)^{11/2}-\frac{17679}{64} (1-2 x)^{9/2}+\frac{17337}{16} (1-2 x)^{7/2}-\frac{832951}{320} (1-2 x)^{5/2}+\frac{381073}{96} (1-2 x)^{3/2}-\frac{290521}{64} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-290521*Sqrt[1 - 2*x])/64 + (381073*(1 - 2*x)^(3/2))/96 - (832951*(1 - 2*x)^(5/2))/320 + (17337*(1 - 2*x)^(7/
2))/16 - (17679*(1 - 2*x)^(9/2))/64 + (13905*(1 - 2*x)^(11/2))/352 - (2025*(1 - 2*x)^(13/2))/832

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}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{290521}{64 \sqrt{1-2 x}}-\frac{381073}{32} \sqrt{1-2 x}+\frac{832951}{64} (1-2 x)^{3/2}-\frac{121359}{16} (1-2 x)^{5/2}+\frac{159111}{64} (1-2 x)^{7/2}-\frac{13905}{32} (1-2 x)^{9/2}+\frac{2025}{64} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac{290521}{64} \sqrt{1-2 x}+\frac{381073}{96} (1-2 x)^{3/2}-\frac{832951}{320} (1-2 x)^{5/2}+\frac{17337}{16} (1-2 x)^{7/2}-\frac{17679}{64} (1-2 x)^{9/2}+\frac{13905}{352} (1-2 x)^{11/2}-\frac{2025}{832} (1-2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0159156, size = 43, normalized size = 0.47 \[ -\frac{\sqrt{1-2 x} \left (334125 x^6+1709100 x^5+3954645 x^4+5576580 x^3+5587044 x^2+4685656 x+4994536\right )}{2145} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(4994536 + 4685656*x + 5587044*x^2 + 5576580*x^3 + 3954645*x^4 + 1709100*x^5 + 334125*x^6))/21
45

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Maple [A]  time = 0.004, size = 40, normalized size = 0.4 \begin{align*} -{\frac{334125\,{x}^{6}+1709100\,{x}^{5}+3954645\,{x}^{4}+5576580\,{x}^{3}+5587044\,{x}^{2}+4685656\,x+4994536}{2145}\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)^(1/2),x)

[Out]

-1/2145*(334125*x^6+1709100*x^5+3954645*x^4+5576580*x^3+5587044*x^2+4685656*x+4994536)*(1-2*x)^(1/2)

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Maxima [A]  time = 2.05838, size = 86, normalized size = 0.93 \begin{align*} -\frac{2025}{832} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{13905}{352} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{17679}{64} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{17337}{16} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{832951}{320} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{381073}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="maxima")

[Out]

-2025/832*(-2*x + 1)^(13/2) + 13905/352*(-2*x + 1)^(11/2) - 17679/64*(-2*x + 1)^(9/2) + 17337/16*(-2*x + 1)^(7
/2) - 832951/320*(-2*x + 1)^(5/2) + 381073/96*(-2*x + 1)^(3/2) - 290521/64*sqrt(-2*x + 1)

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Fricas [A]  time = 1.33756, size = 155, normalized size = 1.68 \begin{align*} -\frac{1}{2145} \,{\left (334125 \, x^{6} + 1709100 \, x^{5} + 3954645 \, x^{4} + 5576580 \, x^{3} + 5587044 \, x^{2} + 4685656 \, x + 4994536\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)^2/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/2145*(334125*x^6 + 1709100*x^5 + 3954645*x^4 + 5576580*x^3 + 5587044*x^2 + 4685656*x + 4994536)*sqrt(-2*x +
1)

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Sympy [A]  time = 54.6966, size = 82, normalized size = 0.89 \begin{align*} - \frac{2025 \left (1 - 2 x\right )^{\frac{13}{2}}}{832} + \frac{13905 \left (1 - 2 x\right )^{\frac{11}{2}}}{352} - \frac{17679 \left (1 - 2 x\right )^{\frac{9}{2}}}{64} + \frac{17337 \left (1 - 2 x\right )^{\frac{7}{2}}}{16} - \frac{832951 \left (1 - 2 x\right )^{\frac{5}{2}}}{320} + \frac{381073 \left (1 - 2 x\right )^{\frac{3}{2}}}{96} - \frac{290521 \sqrt{1 - 2 x}}{64} \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)**(1/2),x)

[Out]

-2025*(1 - 2*x)**(13/2)/832 + 13905*(1 - 2*x)**(11/2)/352 - 17679*(1 - 2*x)**(9/2)/64 + 17337*(1 - 2*x)**(7/2)
/16 - 832951*(1 - 2*x)**(5/2)/320 + 381073*(1 - 2*x)**(3/2)/96 - 290521*sqrt(1 - 2*x)/64

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Giac [A]  time = 1.83958, size = 134, normalized size = 1.46 \begin{align*} -\frac{2025}{832} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{13905}{352} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{17679}{64} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{17337}{16} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{832951}{320} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{381073}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="giac")

[Out]

-2025/832*(2*x - 1)^6*sqrt(-2*x + 1) - 13905/352*(2*x - 1)^5*sqrt(-2*x + 1) - 17679/64*(2*x - 1)^4*sqrt(-2*x +
1) - 17337/16*(2*x - 1)^3*sqrt(-2*x + 1) - 832951/320*(2*x - 1)^2*sqrt(-2*x + 1) + 381073/96*(-2*x + 1)^(3/2)
- 290521/64*sqrt(-2*x + 1)