3.2086 \(\int \frac {(2+3 x)^2 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac {225}{112} (1-2 x)^{7/2}+\frac {51}{2} (1-2 x)^{5/2}-\frac {3467}{24} (1-2 x)^{3/2}+\frac {1309}{2} \sqrt {1-2 x}+\frac {5929}{16 \sqrt {1-2 x}} \]

[Out]

-3467/24*(1-2*x)^(3/2)+51/2*(1-2*x)^(5/2)-225/112*(1-2*x)^(7/2)+5929/16/(1-2*x)^(1/2)+1309/2*(1-2*x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 66, 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} \[ -\frac {225}{112} (1-2 x)^{7/2}+\frac {51}{2} (1-2 x)^{5/2}-\frac {3467}{24} (1-2 x)^{3/2}+\frac {1309}{2} \sqrt {1-2 x}+\frac {5929}{16 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

5929/(16*Sqrt[1 - 2*x]) + (1309*Sqrt[1 - 2*x])/2 - (3467*(1 - 2*x)^(3/2))/24 + (51*(1 - 2*x)^(5/2))/2 - (225*(
1 - 2*x)^(7/2))/112

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)^2}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {5929}{16 (1-2 x)^{3/2}}-\frac {1309}{2 \sqrt {1-2 x}}+\frac {3467}{8} \sqrt {1-2 x}-\frac {255}{2} (1-2 x)^{3/2}+\frac {225}{16} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac {5929}{16 \sqrt {1-2 x}}+\frac {1309}{2} \sqrt {1-2 x}-\frac {3467}{24} (1-2 x)^{3/2}+\frac {51}{2} (1-2 x)^{5/2}-\frac {225}{112} (1-2 x)^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \[ \frac {-675 x^4-2934 x^3-6721 x^2-18230 x+18986}{21 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18986 - 18230*x - 6721*x^2 - 2934*x^3 - 675*x^4)/(21*Sqrt[1 - 2*x])

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fricas [A]  time = 0.82, size = 36, normalized size = 0.55 \[ \frac {{\left (675 \, x^{4} + 2934 \, x^{3} + 6721 \, x^{2} + 18230 \, x - 18986\right )} \sqrt {-2 \, x + 1}}{21 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(675*x^4 + 2934*x^3 + 6721*x^2 + 18230*x - 18986)*sqrt(-2*x + 1)/(2*x - 1)

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giac [A]  time = 1.16, size = 60, normalized size = 0.91 \[ \frac {225}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {51}{2} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {3467}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1309}{2} \, \sqrt {-2 \, x + 1} + \frac {5929}{16 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

225/112*(2*x - 1)^3*sqrt(-2*x + 1) + 51/2*(2*x - 1)^2*sqrt(-2*x + 1) - 3467/24*(-2*x + 1)^(3/2) + 1309/2*sqrt(
-2*x + 1) + 5929/16/sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \[ -\frac {675 x^{4}+2934 x^{3}+6721 x^{2}+18230 x -18986}{21 \sqrt {-2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(675*x^4+2934*x^3+6721*x^2+18230*x-18986)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.49, size = 46, normalized size = 0.70 \[ -\frac {225}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {51}{2} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {3467}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1309}{2} \, \sqrt {-2 \, x + 1} + \frac {5929}{16 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225/112*(-2*x + 1)^(7/2) + 51/2*(-2*x + 1)^(5/2) - 3467/24*(-2*x + 1)^(3/2) + 1309/2*sqrt(-2*x + 1) + 5929/16
/sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 46, normalized size = 0.70 \[ \frac {5929}{16\,\sqrt {1-2\,x}}+\frac {1309\,\sqrt {1-2\,x}}{2}-\frac {3467\,{\left (1-2\,x\right )}^{3/2}}{24}+\frac {51\,{\left (1-2\,x\right )}^{5/2}}{2}-\frac {225\,{\left (1-2\,x\right )}^{7/2}}{112} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5929/(16*(1 - 2*x)^(1/2)) + (1309*(1 - 2*x)^(1/2))/2 - (3467*(1 - 2*x)^(3/2))/24 + (51*(1 - 2*x)^(5/2))/2 - (2
25*(1 - 2*x)^(7/2))/112

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sympy [A]  time = 24.60, size = 58, normalized size = 0.88 \[ - \frac {225 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} + \frac {51 \left (1 - 2 x\right )^{\frac {5}{2}}}{2} - \frac {3467 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} + \frac {1309 \sqrt {1 - 2 x}}{2} + \frac {5929}{16 \sqrt {1 - 2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225*(1 - 2*x)**(7/2)/112 + 51*(1 - 2*x)**(5/2)/2 - 3467*(1 - 2*x)**(3/2)/24 + 1309*sqrt(1 - 2*x)/2 + 5929/(16
*sqrt(1 - 2*x))

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