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

Optimal. Leaf size=92 \[ -\frac {135}{64} (1-2 x)^{9/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {519645}{64} \sqrt {1-2 x}-\frac {60025}{8 \sqrt {1-2 x}}+\frac {184877}{192 (1-2 x)^{3/2}} \]

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} -\frac {135}{64} (1-2 x)^{9/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {519645}{64} \sqrt {1-2 x}-\frac {60025}{8 \sqrt {1-2 x}}+\frac {184877}{192 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

184877/(192*(1 - 2*x)^(3/2)) - 60025/(8*Sqrt[1 - 2*x]) - (519645*Sqrt[1 - 2*x])/64 + (12495*(1 - 2*x)^(3/2))/8
- (19467*(1 - 2*x)^(5/2))/64 + (1053*(1 - 2*x)^(7/2))/28 - (135*(1 - 2*x)^(9/2))/64

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)^5 (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {184877}{64 (1-2 x)^{5/2}}-\frac {60025}{8 (1-2 x)^{3/2}}+\frac {519645}{64 \sqrt {1-2 x}}-\frac {37485}{8} \sqrt {1-2 x}+\frac {97335}{64} (1-2 x)^{3/2}-\frac {1053}{4} (1-2 x)^{5/2}+\frac {1215}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {184877}{192 (1-2 x)^{3/2}}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645}{64} \sqrt {1-2 x}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {135}{64} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {2835 x^6+16767 x^5+49653 x^4+114084 x^3+412812 x^2-844104 x+280696}{21 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(280696 - 844104*x + 412812*x^2 + 114084*x^3 + 49653*x^4 + 16767*x^5 + 2835*x^6)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.03, size = 67, normalized size = 0.73 \begin {gather*} \frac {-2835 (1-2 x)^6+50544 (1-2 x)^5-408807 (1-2 x)^4+2099160 (1-2 x)^3-10912545 (1-2 x)^2-10084200 (1-2 x)+1294139}{1344 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1294139 - 10084200*(1 - 2*x) - 10912545*(1 - 2*x)^2 + 2099160*(1 - 2*x)^3 - 408807*(1 - 2*x)^4 + 50544*(1 - 2
*x)^5 - 2835*(1 - 2*x)^6)/(1344*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.04, size = 51, normalized size = 0.55 \begin {gather*} -\frac {{\left (2835 \, x^{6} + 16767 \, x^{5} + 49653 \, x^{4} + 114084 \, x^{3} + 412812 \, x^{2} - 844104 \, x + 280696\right )} \sqrt {-2 \, x + 1}}{21 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(2835*x^6 + 16767*x^5 + 49653*x^4 + 114084*x^3 + 412812*x^2 - 844104*x + 280696)*sqrt(-2*x + 1)/(4*x^2 -
4*x + 1)

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giac [A]  time = 1.27, size = 88, normalized size = 0.96 \begin {gather*} -\frac {135}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1053}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {19467}{64} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {12495}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {519645}{64} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (1200 \, x - 523\right )}}{192 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135/64*(2*x - 1)^4*sqrt(-2*x + 1) - 1053/28*(2*x - 1)^3*sqrt(-2*x + 1) - 19467/64*(2*x - 1)^2*sqrt(-2*x + 1)
+ 12495/8*(-2*x + 1)^(3/2) - 519645/64*sqrt(-2*x + 1) - 2401/192*(1200*x - 523)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \begin {gather*} -\frac {2835 x^{6}+16767 x^{5}+49653 x^{4}+114084 x^{3}+412812 x^{2}-844104 x +280696}{21 \left (-2 x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(2835*x^6+16767*x^5+49653*x^4+114084*x^3+412812*x^2-844104*x+280696)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.49, size = 60, normalized size = 0.65 \begin {gather*} -\frac {135}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1053}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {19467}{64} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {12495}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {519645}{64} \, \sqrt {-2 \, x + 1} + \frac {2401 \, {\left (1200 \, x - 523\right )}}{192 \, {\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)^5*(3+5*x)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-135/64*(-2*x + 1)^(9/2) + 1053/28*(-2*x + 1)^(7/2) - 19467/64*(-2*x + 1)^(5/2) + 12495/8*(-2*x + 1)^(3/2) - 5
19645/64*sqrt(-2*x + 1) + 2401/192*(1200*x - 523)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 59, normalized size = 0.64 \begin {gather*} \frac {\frac {60025\,x}{4}-\frac {1255723}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {519645\,\sqrt {1-2\,x}}{64}+\frac {12495\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {19467\,{\left (1-2\,x\right )}^{5/2}}{64}+\frac {1053\,{\left (1-2\,x\right )}^{7/2}}{28}-\frac {135\,{\left (1-2\,x\right )}^{9/2}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((60025*x)/4 - 1255723/192)/(1 - 2*x)^(3/2) - (519645*(1 - 2*x)^(1/2))/64 + (12495*(1 - 2*x)^(3/2))/8 - (19467
*(1 - 2*x)^(5/2))/64 + (1053*(1 - 2*x)^(7/2))/28 - (135*(1 - 2*x)^(9/2))/64

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sympy [A]  time = 37.14, size = 82, normalized size = 0.89 \begin {gather*} - \frac {135 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {1053 \left (1 - 2 x\right )^{\frac {7}{2}}}{28} - \frac {19467 \left (1 - 2 x\right )^{\frac {5}{2}}}{64} + \frac {12495 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} - \frac {519645 \sqrt {1 - 2 x}}{64} - \frac {60025}{8 \sqrt {1 - 2 x}} + \frac {184877}{192 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135*(1 - 2*x)**(9/2)/64 + 1053*(1 - 2*x)**(7/2)/28 - 19467*(1 - 2*x)**(5/2)/64 + 12495*(1 - 2*x)**(3/2)/8 - 5
19645*sqrt(1 - 2*x)/64 - 60025/(8*sqrt(1 - 2*x)) + 184877/(192*(1 - 2*x)**(3/2))

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