3.2026 \(\int \frac{(2+3 x)^2 (3+5 x)^3}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=79 \[ \frac{1125}{352} (1-2 x)^{11/2}-\frac{4225}{96} (1-2 x)^{9/2}+\frac{28555}{112} (1-2 x)^{7/2}-\frac{64317}{80} (1-2 x)^{5/2}+\frac{48279}{32} (1-2 x)^{3/2}-\frac{65219}{32} \sqrt{1-2 x} \]

[Out]

(-65219*Sqrt[1 - 2*x])/32 + (48279*(1 - 2*x)^(3/2))/32 - (64317*(1 - 2*x)^(5/2))/80 + (28555*(1 - 2*x)^(7/2))/
112 - (4225*(1 - 2*x)^(9/2))/96 + (1125*(1 - 2*x)^(11/2))/352

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Rubi [A]  time = 0.0149652, 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} \[ \frac{1125}{352} (1-2 x)^{11/2}-\frac{4225}{96} (1-2 x)^{9/2}+\frac{28555}{112} (1-2 x)^{7/2}-\frac{64317}{80} (1-2 x)^{5/2}+\frac{48279}{32} (1-2 x)^{3/2}-\frac{65219}{32} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-65219*Sqrt[1 - 2*x])/32 + (48279*(1 - 2*x)^(3/2))/32 - (64317*(1 - 2*x)^(5/2))/80 + (28555*(1 - 2*x)^(7/2))/
112 - (4225*(1 - 2*x)^(9/2))/96 + (1125*(1 - 2*x)^(11/2))/352

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)^2 (3+5 x)^3}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{65219}{32 \sqrt{1-2 x}}-\frac{144837}{32} \sqrt{1-2 x}+\frac{64317}{16} (1-2 x)^{3/2}-\frac{28555}{16} (1-2 x)^{5/2}+\frac{12675}{32} (1-2 x)^{7/2}-\frac{1125}{32} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{65219}{32} \sqrt{1-2 x}+\frac{48279}{32} (1-2 x)^{3/2}-\frac{64317}{80} (1-2 x)^{5/2}+\frac{28555}{112} (1-2 x)^{7/2}-\frac{4225}{96} (1-2 x)^{9/2}+\frac{1125}{352} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0133192, size = 38, normalized size = 0.48 \[ -\frac{\sqrt{1-2 x} \left (118125 x^5+518000 x^4+1024475 x^3+1252938 x^2+1167932 x+1292672\right )}{1155} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(1292672 + 1167932*x + 1252938*x^2 + 1024475*x^3 + 518000*x^4 + 118125*x^5))/1155

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Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{118125\,{x}^{5}+518000\,{x}^{4}+1024475\,{x}^{3}+1252938\,{x}^{2}+1167932\,x+1292672}{1155}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(118125*x^5+518000*x^4+1024475*x^3+1252938*x^2+1167932*x+1292672)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.07106, size = 74, normalized size = 0.94 \begin{align*} \frac{1125}{352} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{4225}{96} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{28555}{112} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{64317}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{48279}{32} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{65219}{32} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/352*(-2*x + 1)^(11/2) - 4225/96*(-2*x + 1)^(9/2) + 28555/112*(-2*x + 1)^(7/2) - 64317/80*(-2*x + 1)^(5/2)
+ 48279/32*(-2*x + 1)^(3/2) - 65219/32*sqrt(-2*x + 1)

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Fricas [A]  time = 1.65926, size = 135, normalized size = 1.71 \begin{align*} -\frac{1}{1155} \,{\left (118125 \, x^{5} + 518000 \, x^{4} + 1024475 \, x^{3} + 1252938 \, x^{2} + 1167932 \, x + 1292672\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(118125*x^5 + 518000*x^4 + 1024475*x^3 + 1252938*x^2 + 1167932*x + 1292672)*sqrt(-2*x + 1)

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Sympy [A]  time = 42.1917, size = 70, normalized size = 0.89 \begin{align*} \frac{1125 \left (1 - 2 x\right )^{\frac{11}{2}}}{352} - \frac{4225 \left (1 - 2 x\right )^{\frac{9}{2}}}{96} + \frac{28555 \left (1 - 2 x\right )^{\frac{7}{2}}}{112} - \frac{64317 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} + \frac{48279 \left (1 - 2 x\right )^{\frac{3}{2}}}{32} - \frac{65219 \sqrt{1 - 2 x}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125*(1 - 2*x)**(11/2)/352 - 4225*(1 - 2*x)**(9/2)/96 + 28555*(1 - 2*x)**(7/2)/112 - 64317*(1 - 2*x)**(5/2)/80
+ 48279*(1 - 2*x)**(3/2)/32 - 65219*sqrt(1 - 2*x)/32

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Giac [A]  time = 2.51881, size = 112, normalized size = 1.42 \begin{align*} -\frac{1125}{352} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{4225}{96} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{28555}{112} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{64317}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{48279}{32} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{65219}{32} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1125/352*(2*x - 1)^5*sqrt(-2*x + 1) - 4225/96*(2*x - 1)^4*sqrt(-2*x + 1) - 28555/112*(2*x - 1)^3*sqrt(-2*x +
1) - 64317/80*(2*x - 1)^2*sqrt(-2*x + 1) + 48279/32*(-2*x + 1)^(3/2) - 65219/32*sqrt(-2*x + 1)