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

Optimal. Leaf size=92 \[ -\frac {2025}{832} (1-2 x)^{13/2}+\frac {13905}{352} (1-2 x)^{11/2}-\frac {17679}{64} (1-2 x)^{9/2}+\frac {17337}{16} (1-2 x)^{7/2}-\frac {832951}{320} (1-2 x)^{5/2}+\frac {381073}{96} (1-2 x)^{3/2}-\frac {290521}{64} \sqrt {1-2 x} \]

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Rubi [A]  time = 0.02, antiderivative size = 92, 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 {2025}{832} (1-2 x)^{13/2}+\frac {13905}{352} (1-2 x)^{11/2}-\frac {17679}{64} (1-2 x)^{9/2}+\frac {17337}{16} (1-2 x)^{7/2}-\frac {832951}{320} (1-2 x)^{5/2}+\frac {381073}{96} (1-2 x)^{3/2}-\frac {290521}{64} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-290521*Sqrt[1 - 2*x])/64 + (381073*(1 - 2*x)^(3/2))/96 - (832951*(1 - 2*x)^(5/2))/320 + (17337*(1 - 2*x)^(7/
2))/16 - (17679*(1 - 2*x)^(9/2))/64 + (13905*(1 - 2*x)^(11/2))/352 - (2025*(1 - 2*x)^(13/2))/832

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)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {290521}{64 \sqrt {1-2 x}}-\frac {381073}{32} \sqrt {1-2 x}+\frac {832951}{64} (1-2 x)^{3/2}-\frac {121359}{16} (1-2 x)^{5/2}+\frac {159111}{64} (1-2 x)^{7/2}-\frac {13905}{32} (1-2 x)^{9/2}+\frac {2025}{64} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {290521}{64} \sqrt {1-2 x}+\frac {381073}{96} (1-2 x)^{3/2}-\frac {832951}{320} (1-2 x)^{5/2}+\frac {17337}{16} (1-2 x)^{7/2}-\frac {17679}{64} (1-2 x)^{9/2}+\frac {13905}{352} (1-2 x)^{11/2}-\frac {2025}{832} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {1-2 x} \left (334125 x^6+1709100 x^5+3954645 x^4+5576580 x^3+5587044 x^2+4685656 x+4994536\right )}{2145} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2145*(Sqrt[1 - 2*x]*(4994536 + 4685656*x + 5587044*x^2 + 5576580*x^3 + 3954645*x^4 + 1709100*x^5 + 334125*x
^6))

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IntegrateAlgebraic [A]  time = 0.02, size = 82, normalized size = 0.89 \begin {gather*} \frac {-334125 (1-2 x)^{13/2}+5422950 (1-2 x)^{11/2}-37921455 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-357335979 (1-2 x)^{5/2}[phone] (1-2 x)^{3/2}-623167545 \sqrt {1-2 x}}{137280} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-623167545*Sqrt[1 - 2*x] + 544934390*(1 - 2*x)^(3/2) - 357335979*(1 - 2*x)^(5/2) + 148751460*(1 - 2*x)^(7/2)
- 37921455*(1 - 2*x)^(9/2) + 5422950*(1 - 2*x)^(11/2) - 334125*(1 - 2*x)^(13/2))/137280

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fricas [A]  time = 0.84, size = 39, normalized size = 0.42 \begin {gather*} -\frac {1}{2145} \, {\left (334125 \, x^{6} + 1709100 \, x^{5} + 3954645 \, x^{4} + 5576580 \, x^{3} + 5587044 \, x^{2} + 4685656 \, x + 4994536\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)^2/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/2145*(334125*x^6 + 1709100*x^5 + 3954645*x^4 + 5576580*x^3 + 5587044*x^2 + 4685656*x + 4994536)*sqrt(-2*x +
1)

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giac [A]  time = 0.93, size = 99, normalized size = 1.08 \begin {gather*} -\frac {2025}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {13905}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {17679}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {17337}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {832951}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {381073}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {290521}{64} \, \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)^2/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-2025/832*(2*x - 1)^6*sqrt(-2*x + 1) - 13905/352*(2*x - 1)^5*sqrt(-2*x + 1) - 17679/64*(2*x - 1)^4*sqrt(-2*x +
1) - 17337/16*(2*x - 1)^3*sqrt(-2*x + 1) - 832951/320*(2*x - 1)^2*sqrt(-2*x + 1) + 381073/96*(-2*x + 1)^(3/2)
- 290521/64*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \begin {gather*} -\frac {\left (334125 x^{6}+1709100 x^{5}+3954645 x^{4}+5576580 x^{3}+5587044 x^{2}+4685656 x +4994536\right ) \sqrt {-2 x +1}}{2145} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2145*(334125*x^6+1709100*x^5+3954645*x^4+5576580*x^3+5587044*x^2+4685656*x+4994536)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.43, size = 64, normalized size = 0.70 \begin {gather*} -\frac {2025}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {13905}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {17679}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {17337}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {832951}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {381073}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {290521}{64} \, \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)^2/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-2025/832*(-2*x + 1)^(13/2) + 13905/352*(-2*x + 1)^(11/2) - 17679/64*(-2*x + 1)^(9/2) + 17337/16*(-2*x + 1)^(7
/2) - 832951/320*(-2*x + 1)^(5/2) + 381073/96*(-2*x + 1)^(3/2) - 290521/64*sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {381073\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {290521\,\sqrt {1-2\,x}}{64}-\frac {832951\,{\left (1-2\,x\right )}^{5/2}}{320}+\frac {17337\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {17679\,{\left (1-2\,x\right )}^{9/2}}{64}+\frac {13905\,{\left (1-2\,x\right )}^{11/2}}{352}-\frac {2025\,{\left (1-2\,x\right )}^{13/2}}{832} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(381073*(1 - 2*x)^(3/2))/96 - (290521*(1 - 2*x)^(1/2))/64 - (832951*(1 - 2*x)^(5/2))/320 + (17337*(1 - 2*x)^(7
/2))/16 - (17679*(1 - 2*x)^(9/2))/64 + (13905*(1 - 2*x)^(11/2))/352 - (2025*(1 - 2*x)^(13/2))/832

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sympy [A]  time = 89.64, size = 82, normalized size = 0.89 \begin {gather*} - \frac {2025 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {13905 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} - \frac {17679 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {17337 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} - \frac {832951 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} + \frac {381073 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} - \frac {290521 \sqrt {1 - 2 x}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2025*(1 - 2*x)**(13/2)/832 + 13905*(1 - 2*x)**(11/2)/352 - 17679*(1 - 2*x)**(9/2)/64 + 17337*(1 - 2*x)**(7/2)
/16 - 832951*(1 - 2*x)**(5/2)/320 + 381073*(1 - 2*x)**(3/2)/96 - 290521*sqrt(1 - 2*x)/64

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