3.2001 \(\int \frac{(2+3 x)^5 (3+5 x)}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{1215}{832} (1-2 x)^{13/2}+\frac{1053}{44} (1-2 x)^{11/2}-\frac{10815}{64} (1-2 x)^{9/2}+\frac{5355}{8} (1-2 x)^{7/2}-\frac{103929}{64} (1-2 x)^{5/2}+\frac{60025}{24} (1-2 x)^{3/2}-\frac{184877}{64} \sqrt{1-2 x} \]

[Out]

(-184877*Sqrt[1 - 2*x])/64 + (60025*(1 - 2*x)^(3/2))/24 - (103929*(1 - 2*x)^(5/2))/64 + (5355*(1 - 2*x)^(7/2))
/8 - (10815*(1 - 2*x)^(9/2))/64 + (1053*(1 - 2*x)^(11/2))/44 - (1215*(1 - 2*x)^(13/2))/832

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Rubi [A]  time = 0.0167062, antiderivative size = 92, normalized size of antiderivative = 1., 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} \[ -\frac{1215}{832} (1-2 x)^{13/2}+\frac{1053}{44} (1-2 x)^{11/2}-\frac{10815}{64} (1-2 x)^{9/2}+\frac{5355}{8} (1-2 x)^{7/2}-\frac{103929}{64} (1-2 x)^{5/2}+\frac{60025}{24} (1-2 x)^{3/2}-\frac{184877}{64} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-184877*Sqrt[1 - 2*x])/64 + (60025*(1 - 2*x)^(3/2))/24 - (103929*(1 - 2*x)^(5/2))/64 + (5355*(1 - 2*x)^(7/2))
/8 - (10815*(1 - 2*x)^(9/2))/64 + (1053*(1 - 2*x)^(11/2))/44 - (1215*(1 - 2*x)^(13/2))/832

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)}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{184877}{64 \sqrt{1-2 x}}-\frac{60025}{8} \sqrt{1-2 x}+\frac{519645}{64} (1-2 x)^{3/2}-\frac{37485}{8} (1-2 x)^{5/2}+\frac{97335}{64} (1-2 x)^{7/2}-\frac{1053}{4} (1-2 x)^{9/2}+\frac{1215}{64} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac{184877}{64} \sqrt{1-2 x}+\frac{60025}{24} (1-2 x)^{3/2}-\frac{103929}{64} (1-2 x)^{5/2}+\frac{5355}{8} (1-2 x)^{7/2}-\frac{10815}{64} (1-2 x)^{9/2}+\frac{1053}{44} (1-2 x)^{11/2}-\frac{1215}{832} (1-2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.016173, size = 43, normalized size = 0.47 \[ -\frac{1}{429} \sqrt{1-2 x} \left (40095 x^6+208251 x^5+488925 x^4+698580 x^3+707436 x^2+597464 x+638648\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(638648 + 597464*x + 707436*x^2 + 698580*x^3 + 488925*x^4 + 208251*x^5 + 40095*x^6))/429

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Maple [A]  time = 0.004, size = 40, normalized size = 0.4 \begin{align*} -{\frac{40095\,{x}^{6}+208251\,{x}^{5}+488925\,{x}^{4}+698580\,{x}^{3}+707436\,{x}^{2}+597464\,x+638648}{429}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/429*(40095*x^6+208251*x^5+488925*x^4+698580*x^3+707436*x^2+597464*x+638648)*(1-2*x)^(1/2)

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Maxima [A]  time = 2.37661, size = 86, normalized size = 0.93 \begin{align*} -\frac{1215}{832} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{1053}{44} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{10815}{64} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{5355}{8} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{103929}{64} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{60025}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{184877}{64} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215/832*(-2*x + 1)^(13/2) + 1053/44*(-2*x + 1)^(11/2) - 10815/64*(-2*x + 1)^(9/2) + 5355/8*(-2*x + 1)^(7/2)
- 103929/64*(-2*x + 1)^(5/2) + 60025/24*(-2*x + 1)^(3/2) - 184877/64*sqrt(-2*x + 1)

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Fricas [A]  time = 1.42338, size = 144, normalized size = 1.57 \begin{align*} -\frac{1}{429} \,{\left (40095 \, x^{6} + 208251 \, x^{5} + 488925 \, x^{4} + 698580 \, x^{3} + 707436 \, x^{2} + 597464 \, x + 638648\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/429*(40095*x^6 + 208251*x^5 + 488925*x^4 + 698580*x^3 + 707436*x^2 + 597464*x + 638648)*sqrt(-2*x + 1)

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Sympy [A]  time = 54.7853, size = 82, normalized size = 0.89 \begin{align*} - \frac{1215 \left (1 - 2 x\right )^{\frac{13}{2}}}{832} + \frac{1053 \left (1 - 2 x\right )^{\frac{11}{2}}}{44} - \frac{10815 \left (1 - 2 x\right )^{\frac{9}{2}}}{64} + \frac{5355 \left (1 - 2 x\right )^{\frac{7}{2}}}{8} - \frac{103929 \left (1 - 2 x\right )^{\frac{5}{2}}}{64} + \frac{60025 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} - \frac{184877 \sqrt{1 - 2 x}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215*(1 - 2*x)**(13/2)/832 + 1053*(1 - 2*x)**(11/2)/44 - 10815*(1 - 2*x)**(9/2)/64 + 5355*(1 - 2*x)**(7/2)/8
- 103929*(1 - 2*x)**(5/2)/64 + 60025*(1 - 2*x)**(3/2)/24 - 184877*sqrt(1 - 2*x)/64

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Giac [A]  time = 2.31124, size = 134, normalized size = 1.46 \begin{align*} -\frac{1215}{832} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{1053}{44} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{10815}{64} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{5355}{8} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{103929}{64} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{60025}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{184877}{64} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215/832*(2*x - 1)^6*sqrt(-2*x + 1) - 1053/44*(2*x - 1)^5*sqrt(-2*x + 1) - 10815/64*(2*x - 1)^4*sqrt(-2*x + 1
) - 5355/8*(2*x - 1)^3*sqrt(-2*x + 1) - 103929/64*(2*x - 1)^2*sqrt(-2*x + 1) + 60025/24*(-2*x + 1)^(3/2) - 184
877/64*sqrt(-2*x + 1)