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

Optimal. Leaf size=79 \[ \frac {45}{32} (1-2 x)^{9/2}-\frac {4671}{224} (1-2 x)^{7/2}+\frac {10773}{80} (1-2 x)^{5/2}-\frac {8281}{16} (1-2 x)^{3/2}+\frac {57281}{32} \sqrt {1-2 x}+\frac {26411}{32 \sqrt {1-2 x}} \]

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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 {45}{32} (1-2 x)^{9/2}-\frac {4671}{224} (1-2 x)^{7/2}+\frac {10773}{80} (1-2 x)^{5/2}-\frac {8281}{16} (1-2 x)^{3/2}+\frac {57281}{32} \sqrt {1-2 x}+\frac {26411}{32 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

26411/(32*Sqrt[1 - 2*x]) + (57281*Sqrt[1 - 2*x])/32 - (8281*(1 - 2*x)^(3/2))/16 + (10773*(1 - 2*x)^(5/2))/80 -
(4671*(1 - 2*x)^(7/2))/224 + (45*(1 - 2*x)^(9/2))/32

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)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {26411}{32 (1-2 x)^{3/2}}-\frac {57281}{32 \sqrt {1-2 x}}+\frac {24843}{16} \sqrt {1-2 x}-\frac {10773}{16} (1-2 x)^{3/2}+\frac {4671}{32} (1-2 x)^{5/2}-\frac {405}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {26411}{32 \sqrt {1-2 x}}+\frac {57281}{32} \sqrt {1-2 x}-\frac {8281}{16} (1-2 x)^{3/2}+\frac {10773}{80} (1-2 x)^{5/2}-\frac {4671}{224} (1-2 x)^{7/2}+\frac {45}{32} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} \frac {-1575 x^5-7740 x^4-18288 x^3-31448 x^2-75776 x+77456}{35 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(77456 - 75776*x - 31448*x^2 - 18288*x^3 - 7740*x^4 - 1575*x^5)/(35*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.02, size = 58, normalized size = 0.73 \begin {gather*} \frac {1575 (1-2 x)^5-23355 (1-2 x)^4+150822 (1-2 x)^3-579670 (1-2 x)^2+2004835 (1-2 x)+924385}{1120 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(924385 + 2004835*(1 - 2*x) - 579670*(1 - 2*x)^2 + 150822*(1 - 2*x)^3 - 23355*(1 - 2*x)^4 + 1575*(1 - 2*x)^5)/
(1120*Sqrt[1 - 2*x])

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fricas [A]  time = 0.98, size = 41, normalized size = 0.52 \begin {gather*} \frac {{\left (1575 \, x^{5} + 7740 \, x^{4} + 18288 \, x^{3} + 31448 \, x^{2} + 75776 \, x - 77456\right )} \sqrt {-2 \, x + 1}}{35 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(1575*x^5 + 7740*x^4 + 18288*x^3 + 31448*x^2 + 75776*x - 77456)*sqrt(-2*x + 1)/(2*x - 1)

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giac [A]  time = 1.28, size = 76, normalized size = 0.96 \begin {gather*} \frac {45}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {4671}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {10773}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {8281}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {57281}{32} \, \sqrt {-2 \, x + 1} + \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)^(3/2),x, algorithm="giac")

[Out]

45/32*(2*x - 1)^4*sqrt(-2*x + 1) + 4671/224*(2*x - 1)^3*sqrt(-2*x + 1) + 10773/80*(2*x - 1)^2*sqrt(-2*x + 1) -
8281/16*(-2*x + 1)^(3/2) + 57281/32*sqrt(-2*x + 1) + 26411/32/sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {1575 x^{5}+7740 x^{4}+18288 x^{3}+31448 x^{2}+75776 x -77456}{35 \sqrt {-2 x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(1575*x^5+7740*x^4+18288*x^3+31448*x^2+75776*x-77456)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.58, size = 55, normalized size = 0.70 \begin {gather*} \frac {45}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {4671}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {10773}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {8281}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {57281}{32} \, \sqrt {-2 \, x + 1} + \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)^(3/2),x, algorithm="maxima")

[Out]

45/32*(-2*x + 1)^(9/2) - 4671/224*(-2*x + 1)^(7/2) + 10773/80*(-2*x + 1)^(5/2) - 8281/16*(-2*x + 1)^(3/2) + 57
281/32*sqrt(-2*x + 1) + 26411/32/sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {26411}{32\,\sqrt {1-2\,x}}+\frac {57281\,\sqrt {1-2\,x}}{32}-\frac {8281\,{\left (1-2\,x\right )}^{3/2}}{16}+\frac {10773\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {4671\,{\left (1-2\,x\right )}^{7/2}}{224}+\frac {45\,{\left (1-2\,x\right )}^{9/2}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

26411/(32*(1 - 2*x)^(1/2)) + (57281*(1 - 2*x)^(1/2))/32 - (8281*(1 - 2*x)^(3/2))/16 + (10773*(1 - 2*x)^(5/2))/
80 - (4671*(1 - 2*x)^(7/2))/224 + (45*(1 - 2*x)^(9/2))/32

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sympy [A]  time = 32.81, size = 70, normalized size = 0.89 \begin {gather*} \frac {45 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {4671 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} + \frac {10773 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {8281 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} + \frac {57281 \sqrt {1 - 2 x}}{32} + \frac {26411}{32 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

45*(1 - 2*x)**(9/2)/32 - 4671*(1 - 2*x)**(7/2)/224 + 10773*(1 - 2*x)**(5/2)/80 - 8281*(1 - 2*x)**(3/2)/16 + 57
281*sqrt(1 - 2*x)/32 + 26411/(32*sqrt(1 - 2*x))

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