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

Optimal. Leaf size=66 \[ -\frac {25}{16} (1-2 x)^{9/2}+\frac {255}{14} (1-2 x)^{7/2}-\frac {3467}{40} (1-2 x)^{5/2}+\frac {1309}{6} (1-2 x)^{3/2}-\frac {5929}{16} \sqrt {1-2 x} \]

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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} \begin {gather*} -\frac {25}{16} (1-2 x)^{9/2}+\frac {255}{14} (1-2 x)^{7/2}-\frac {3467}{40} (1-2 x)^{5/2}+\frac {1309}{6} (1-2 x)^{3/2}-\frac {5929}{16} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5929*Sqrt[1 - 2*x])/16 + (1309*(1 - 2*x)^(3/2))/6 - (3467*(1 - 2*x)^(5/2))/40 + (255*(1 - 2*x)^(7/2))/14 - (
25*(1 - 2*x)^(9/2))/16

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}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {5929}{16 \sqrt {1-2 x}}-\frac {1309}{2} \sqrt {1-2 x}+\frac {3467}{8} (1-2 x)^{3/2}-\frac {255}{2} (1-2 x)^{5/2}+\frac {225}{16} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {5929}{16} \sqrt {1-2 x}+\frac {1309}{6} (1-2 x)^{3/2}-\frac {3467}{40} (1-2 x)^{5/2}+\frac {255}{14} (1-2 x)^{7/2}-\frac {25}{16} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {1}{105} \sqrt {1-2 x} \left (2625 x^4+10050 x^3+17391 x^2+19574 x+23354\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(Sqrt[1 - 2*x]*(23354 + 19574*x + 17391*x^2 + 10050*x^3 + 2625*x^4))

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IntegrateAlgebraic [A]  time = 0.02, size = 60, normalized size = 0.91 \begin {gather*} \frac {-2625 (1-2 x)^{9/2}+30600 (1-2 x)^{7/2}-145614 (1-2 x)^{5/2}+366520 (1-2 x)^{3/2}-622545 \sqrt {1-2 x}}{1680} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-622545*Sqrt[1 - 2*x] + 366520*(1 - 2*x)^(3/2) - 145614*(1 - 2*x)^(5/2) + 30600*(1 - 2*x)^(7/2) - 2625*(1 - 2
*x)^(9/2))/1680

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fricas [A]  time = 1.42, size = 29, normalized size = 0.44 \begin {gather*} -\frac {1}{105} \, {\left (2625 \, x^{4} + 10050 \, x^{3} + 17391 \, x^{2} + 19574 \, x + 23354\right )} \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)^2/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/105*(2625*x^4 + 10050*x^3 + 17391*x^2 + 19574*x + 23354)*sqrt(-2*x + 1)

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giac [A]  time = 0.96, size = 67, normalized size = 1.02 \begin {gather*} -\frac {25}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {255}{14} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {3467}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1309}{6} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {5929}{16} \, \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)^2/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-25/16*(2*x - 1)^4*sqrt(-2*x + 1) - 255/14*(2*x - 1)^3*sqrt(-2*x + 1) - 3467/40*(2*x - 1)^2*sqrt(-2*x + 1) + 1
309/6*(-2*x + 1)^(3/2) - 5929/16*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 30, normalized size = 0.45 \begin {gather*} -\frac {\left (2625 x^{4}+10050 x^{3}+17391 x^{2}+19574 x +23354\right ) \sqrt {-2 x +1}}{105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(2625*x^4+10050*x^3+17391*x^2+19574*x+23354)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.59, size = 46, normalized size = 0.70 \begin {gather*} -\frac {25}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {255}{14} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {3467}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1309}{6} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {5929}{16} \, \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)^2/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-25/16*(-2*x + 1)^(9/2) + 255/14*(-2*x + 1)^(7/2) - 3467/40*(-2*x + 1)^(5/2) + 1309/6*(-2*x + 1)^(3/2) - 5929/
16*sqrt(-2*x + 1)

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mupad [B]  time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {1309\,{\left (1-2\,x\right )}^{3/2}}{6}-\frac {5929\,\sqrt {1-2\,x}}{16}-\frac {3467\,{\left (1-2\,x\right )}^{5/2}}{40}+\frac {255\,{\left (1-2\,x\right )}^{7/2}}{14}-\frac {25\,{\left (1-2\,x\right )}^{9/2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1309*(1 - 2*x)^(3/2))/6 - (5929*(1 - 2*x)^(1/2))/16 - (3467*(1 - 2*x)^(5/2))/40 + (255*(1 - 2*x)^(7/2))/14 -
(25*(1 - 2*x)^(9/2))/16

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sympy [A]  time = 47.64, size = 58, normalized size = 0.88 \begin {gather*} - \frac {25 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} + \frac {255 \left (1 - 2 x\right )^{\frac {7}{2}}}{14} - \frac {3467 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} + \frac {1309 \left (1 - 2 x\right )^{\frac {3}{2}}}{6} - \frac {5929 \sqrt {1 - 2 x}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25*(1 - 2*x)**(9/2)/16 + 255*(1 - 2*x)**(7/2)/14 - 3467*(1 - 2*x)**(5/2)/40 + 1309*(1 - 2*x)**(3/2)/6 - 5929*
sqrt(1 - 2*x)/16

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