3.19.63 \(\int \frac {(2+3 x)^4 (3+5 x)}{\sqrt {1-2 x}} \, dx\)

Optimal. Leaf size=79 \[ \frac {405}{352} (1-2 x)^{11/2}-\frac {519}{32} (1-2 x)^{9/2}+\frac {1539}{16} (1-2 x)^{7/2}-\frac {24843}{80} (1-2 x)^{5/2}+\frac {57281}{96} (1-2 x)^{3/2}-\frac {26411}{32} \sqrt {1-2 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} \frac {405}{352} (1-2 x)^{11/2}-\frac {519}{32} (1-2 x)^{9/2}+\frac {1539}{16} (1-2 x)^{7/2}-\frac {24843}{80} (1-2 x)^{5/2}+\frac {57281}{96} (1-2 x)^{3/2}-\frac {26411}{32} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^4*(3 + 5*x))/Sqrt[1 - 2*x],x]

[Out]

(-26411*Sqrt[1 - 2*x])/32 + (57281*(1 - 2*x)^(3/2))/96 - (24843*(1 - 2*x)^(5/2))/80 + (1539*(1 - 2*x)^(7/2))/1
6 - (519*(1 - 2*x)^(9/2))/32 + (405*(1 - 2*x)^(11/2))/352

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {26411}{32 \sqrt {1-2 x}}-\frac {57281}{32} \sqrt {1-2 x}+\frac {24843}{16} (1-2 x)^{3/2}-\frac {10773}{16} (1-2 x)^{5/2}+\frac {4671}{32} (1-2 x)^{7/2}-\frac {405}{32} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {26411}{32} \sqrt {1-2 x}+\frac {57281}{96} (1-2 x)^{3/2}-\frac {24843}{80} (1-2 x)^{5/2}+\frac {1539}{16} (1-2 x)^{7/2}-\frac {519}{32} (1-2 x)^{9/2}+\frac {405}{352} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {1}{165} \sqrt {1-2 x} \left (6075 x^5+27630 x^4+56520 x^3+71136 x^2+67664 x+75584\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^4*(3 + 5*x))/Sqrt[1 - 2*x],x]

[Out]

-1/165*(Sqrt[1 - 2*x]*(75584 + 67664*x + 71136*x^2 + 56520*x^3 + 27630*x^4 + 6075*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 58, normalized size = 0.73 \begin {gather*} \frac {\left (6075 (1-2 x)^5-85635 (1-2 x)^4+507870 (1-2 x)^3-1639638 (1-2 x)^2+3150455 (1-2 x)-4357815\right ) \sqrt {1-2 x}}{5280} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + 3*x)^4*(3 + 5*x))/Sqrt[1 - 2*x],x]

[Out]

((-4357815 + 3150455*(1 - 2*x) - 1639638*(1 - 2*x)^2 + 507870*(1 - 2*x)^3 - 85635*(1 - 2*x)^4 + 6075*(1 - 2*x)
^5)*Sqrt[1 - 2*x])/5280

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fricas [A]  time = 1.07, size = 34, normalized size = 0.43 \begin {gather*} -\frac {1}{165} \, {\left (6075 \, x^{5} + 27630 \, x^{4} + 56520 \, x^{3} + 71136 \, x^{2} + 67664 \, x + 75584\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/165*(6075*x^5 + 27630*x^4 + 56520*x^3 + 71136*x^2 + 67664*x + 75584)*sqrt(-2*x + 1)

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giac [A]  time = 0.98, size = 83, normalized size = 1.05 \begin {gather*} -\frac {405}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {519}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1539}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {24843}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {57281}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {26411}{32} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-405/352*(2*x - 1)^5*sqrt(-2*x + 1) - 519/32*(2*x - 1)^4*sqrt(-2*x + 1) - 1539/16*(2*x - 1)^3*sqrt(-2*x + 1) -
24843/80*(2*x - 1)^2*sqrt(-2*x + 1) + 57281/96*(-2*x + 1)^(3/2) - 26411/32*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (6075 x^{5}+27630 x^{4}+56520 x^{3}+71136 x^{2}+67664 x +75584\right ) \sqrt {-2 x +1}}{165} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^4*(5*x+3)/(-2*x+1)^(1/2),x)

[Out]

-1/165*(6075*x^5+27630*x^4+56520*x^3+71136*x^2+67664*x+75584)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.48, size = 55, normalized size = 0.70 \begin {gather*} \frac {405}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {519}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1539}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {24843}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {57281}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {26411}{32} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

405/352*(-2*x + 1)^(11/2) - 519/32*(-2*x + 1)^(9/2) + 1539/16*(-2*x + 1)^(7/2) - 24843/80*(-2*x + 1)^(5/2) + 5
7281/96*(-2*x + 1)^(3/2) - 26411/32*sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {57281\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {26411\,\sqrt {1-2\,x}}{32}-\frac {24843\,{\left (1-2\,x\right )}^{5/2}}{80}+\frac {1539\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {519\,{\left (1-2\,x\right )}^{9/2}}{32}+\frac {405\,{\left (1-2\,x\right )}^{11/2}}{352} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x + 2)^4*(5*x + 3))/(1 - 2*x)^(1/2),x)

[Out]

(57281*(1 - 2*x)^(3/2))/96 - (26411*(1 - 2*x)^(1/2))/32 - (24843*(1 - 2*x)^(5/2))/80 + (1539*(1 - 2*x)^(7/2))/
16 - (519*(1 - 2*x)^(9/2))/32 + (405*(1 - 2*x)^(11/2))/352

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sympy [A]  time = 61.45, size = 70, normalized size = 0.89 \begin {gather*} \frac {405 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} - \frac {519 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} + \frac {1539 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} - \frac {24843 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} + \frac {57281 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} - \frac {26411 \sqrt {1 - 2 x}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4*(3+5*x)/(1-2*x)**(1/2),x)

[Out]

405*(1 - 2*x)**(11/2)/352 - 519*(1 - 2*x)**(9/2)/32 + 1539*(1 - 2*x)**(7/2)/16 - 24843*(1 - 2*x)**(5/2)/80 + 5
7281*(1 - 2*x)**(3/2)/96 - 26411*sqrt(1 - 2*x)/32

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