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

Optimal. Leaf size=79 \[ \frac {125}{32} (1-2 x)^{9/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {144837}{32} \sqrt {1-2 x}+\frac {65219}{32 \sqrt {1-2 x}} \]

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Rubi [A]  time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {125}{32} (1-2 x)^{9/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {144837}{32} \sqrt {1-2 x}+\frac {65219}{32 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

65219/(32*Sqrt[1 - 2*x]) + (144837*Sqrt[1 - 2*x])/32 - (21439*(1 - 2*x)^(3/2))/16 + (5711*(1 - 2*x)^(5/2))/16
- (12675*(1 - 2*x)^(7/2))/224 + (125*(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)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {65219}{32 (1-2 x)^{3/2}}-\frac {144837}{32 \sqrt {1-2 x}}+\frac {64317}{16} \sqrt {1-2 x}-\frac {28555}{16} (1-2 x)^{3/2}+\frac {12675}{32} (1-2 x)^{5/2}-\frac {1125}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {65219}{32 \sqrt {1-2 x}}+\frac {144837}{32} \sqrt {1-2 x}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {125}{32} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 38, normalized size = 0.48 \begin {gather*} \frac {-875 x^5-4150 x^4-9501 x^3-15948 x^2-37944 x+38700}{7 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(38700 - 37944*x - 15948*x^2 - 9501*x^3 - 4150*x^4 - 875*x^5)/(7*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.02, size = 58, normalized size = 0.73 \begin {gather*} \frac {875 (1-2 x)^5-12675 (1-2 x)^4+79954 (1-2 x)^3-300146 (1-2 x)^2+1013859 (1-2 x)+456533}{224 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(456533 + 1013859*(1 - 2*x) - 300146*(1 - 2*x)^2 + 79954*(1 - 2*x)^3 - 12675*(1 - 2*x)^4 + 875*(1 - 2*x)^5)/(2
24*Sqrt[1 - 2*x])

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fricas [A]  time = 1.62, size = 41, normalized size = 0.52 \begin {gather*} \frac {{\left (875 \, x^{5} + 4150 \, x^{4} + 9501 \, x^{3} + 15948 \, x^{2} + 37944 \, x - 38700\right )} \sqrt {-2 \, x + 1}}{7 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(875*x^5 + 4150*x^4 + 9501*x^3 + 15948*x^2 + 37944*x - 38700)*sqrt(-2*x + 1)/(2*x - 1)

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giac [A]  time = 1.26, size = 76, normalized size = 0.96 \begin {gather*} \frac {125}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {12675}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {5711}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {21439}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {144837}{32} \, \sqrt {-2 \, x + 1} + \frac {65219}{32 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/32*(2*x - 1)^4*sqrt(-2*x + 1) + 12675/224*(2*x - 1)^3*sqrt(-2*x + 1) + 5711/16*(2*x - 1)^2*sqrt(-2*x + 1)
- 21439/16*(-2*x + 1)^(3/2) + 144837/32*sqrt(-2*x + 1) + 65219/32/sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {875 x^{5}+4150 x^{4}+9501 x^{3}+15948 x^{2}+37944 x -38700}{7 \sqrt {-2 x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(875*x^5+4150*x^4+9501*x^3+15948*x^2+37944*x-38700)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.49, size = 55, normalized size = 0.70 \begin {gather*} \frac {125}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {12675}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {5711}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {21439}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {144837}{32} \, \sqrt {-2 \, x + 1} + \frac {65219}{32 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/32*(-2*x + 1)^(9/2) - 12675/224*(-2*x + 1)^(7/2) + 5711/16*(-2*x + 1)^(5/2) - 21439/16*(-2*x + 1)^(3/2) +
144837/32*sqrt(-2*x + 1) + 65219/32/sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {65219}{32\,\sqrt {1-2\,x}}+\frac {144837\,\sqrt {1-2\,x}}{32}-\frac {21439\,{\left (1-2\,x\right )}^{3/2}}{16}+\frac {5711\,{\left (1-2\,x\right )}^{5/2}}{16}-\frac {12675\,{\left (1-2\,x\right )}^{7/2}}{224}+\frac {125\,{\left (1-2\,x\right )}^{9/2}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

65219/(32*(1 - 2*x)^(1/2)) + (144837*(1 - 2*x)^(1/2))/32 - (21439*(1 - 2*x)^(3/2))/16 + (5711*(1 - 2*x)^(5/2))
/16 - (12675*(1 - 2*x)^(7/2))/224 + (125*(1 - 2*x)^(9/2))/32

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sympy [A]  time = 32.66, size = 70, normalized size = 0.89 \begin {gather*} \frac {125 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {12675 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} + \frac {5711 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} - \frac {21439 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} + \frac {144837 \sqrt {1 - 2 x}}{32} + \frac {65219}{32 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*(1 - 2*x)**(9/2)/32 - 12675*(1 - 2*x)**(7/2)/224 + 5711*(1 - 2*x)**(5/2)/16 - 21439*(1 - 2*x)**(3/2)/16 +
144837*sqrt(1 - 2*x)/32 + 65219/(32*sqrt(1 - 2*x))

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