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

Optimal. Leaf size=79 \[ \frac {675}{416} (1-2 x)^{13/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {41503}{96} (1-2 x)^{3/2} \]

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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 {675}{416} (1-2 x)^{13/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {41503}{96} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41503*(1 - 2*x)^(3/2))/96 + (91091*(1 - 2*x)^(5/2))/160 - (5711*(1 - 2*x)^(7/2))/16 + (1949*(1 - 2*x)^(9/2))
/16 - (7695*(1 - 2*x)^(11/2))/352 + (675*(1 - 2*x)^(13/2))/416

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 \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx &=\int \left (\frac {41503}{32} \sqrt {1-2 x}-\frac {91091}{32} (1-2 x)^{3/2}+\frac {39977}{16} (1-2 x)^{5/2}-\frac {17541}{16} (1-2 x)^{7/2}+\frac {7695}{32} (1-2 x)^{9/2}-\frac {675}{32} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {41503}{96} (1-2 x)^{3/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {675}{416} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(1-2 x)^{3/2} \left (111375 x^5+471825 x^4+868215 x^3+913245 x^2+607254 x+253898\right )}{2145} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2145*((1 - 2*x)^(3/2)*(253898 + 607254*x + 913245*x^2 + 868215*x^3 + 471825*x^4 + 111375*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 71, normalized size = 0.90 \begin {gather*} \frac {111375 (1-2 x)^{13/2}-1500525 (1-2 x)^{11/2}+8361210 (1-2 x)^{9/2}-24500190 (1-2 x)^{7/2}[phone] (1-2 x)^{5/2}-29674645 (1-2 x)^{3/2}}{68640} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-29674645*(1 - 2*x)^(3/2) + 39078039*(1 - 2*x)^(5/2) - 24500190*(1 - 2*x)^(7/2) + 8361210*(1 - 2*x)^(9/2) - 1
500525*(1 - 2*x)^(11/2) + 111375*(1 - 2*x)^(13/2))/68640

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fricas [A]  time = 1.15, size = 39, normalized size = 0.49 \begin {gather*} \frac {1}{2145} \, {\left (222750 \, x^{6} + 832275 \, x^{5} + 1264605 \, x^{4} + 958275 \, x^{3} + 301263 \, x^{2} - 99458 \, x - 253898\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2145*(222750*x^6 + 832275*x^5 + 1264605*x^4 + 958275*x^3 + 301263*x^2 - 99458*x - 253898)*sqrt(-2*x + 1)

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giac [A]  time = 1.29, size = 90, normalized size = 1.14 \begin {gather*} \frac {675}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {7695}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1949}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {5711}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {91091}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {41503}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675/416*(2*x - 1)^6*sqrt(-2*x + 1) + 7695/352*(2*x - 1)^5*sqrt(-2*x + 1) + 1949/16*(2*x - 1)^4*sqrt(-2*x + 1)
+ 5711/16*(2*x - 1)^3*sqrt(-2*x + 1) + 91091/160*(2*x - 1)^2*sqrt(-2*x + 1) - 41503/96*(-2*x + 1)^(3/2)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (111375 x^{5}+471825 x^{4}+868215 x^{3}+913245 x^{2}+607254 x +253898\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{2145} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2145*(111375*x^5+471825*x^4+868215*x^3+913245*x^2+607254*x+253898)*(-2*x+1)^(3/2)

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maxima [A]  time = 0.69, size = 55, normalized size = 0.70 \begin {gather*} \frac {675}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {7695}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1949}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {5711}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {91091}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {41503}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675/416*(-2*x + 1)^(13/2) - 7695/352*(-2*x + 1)^(11/2) + 1949/16*(-2*x + 1)^(9/2) - 5711/16*(-2*x + 1)^(7/2) +
91091/160*(-2*x + 1)^(5/2) - 41503/96*(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {91091\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {41503\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {5711\,{\left (1-2\,x\right )}^{7/2}}{16}+\frac {1949\,{\left (1-2\,x\right )}^{9/2}}{16}-\frac {7695\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {675\,{\left (1-2\,x\right )}^{13/2}}{416} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(91091*(1 - 2*x)^(5/2))/160 - (41503*(1 - 2*x)^(3/2))/96 - (5711*(1 - 2*x)^(7/2))/16 + (1949*(1 - 2*x)^(9/2))/
16 - (7695*(1 - 2*x)^(11/2))/352 + (675*(1 - 2*x)^(13/2))/416

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sympy [A]  time = 2.81, size = 70, normalized size = 0.89 \begin {gather*} \frac {675 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {7695 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {1949 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} - \frac {5711 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} + \frac {91091 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {41503 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675*(1 - 2*x)**(13/2)/416 - 7695*(1 - 2*x)**(11/2)/352 + 1949*(1 - 2*x)**(9/2)/16 - 5711*(1 - 2*x)**(7/2)/16 +
91091*(1 - 2*x)**(5/2)/160 - 41503*(1 - 2*x)**(3/2)/96

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