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

Optimal. Leaf size=53 \[ \frac {75}{56} (1-2 x)^{7/2}-\frac {101}{8} (1-2 x)^{5/2}+\frac {1133}{24} (1-2 x)^{3/2}-\frac {847}{8} \sqrt {1-2 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 53, 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 {75}{56} (1-2 x)^{7/2}-\frac {101}{8} (1-2 x)^{5/2}+\frac {1133}{24} (1-2 x)^{3/2}-\frac {847}{8} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-847*Sqrt[1 - 2*x])/8 + (1133*(1 - 2*x)^(3/2))/24 - (101*(1 - 2*x)^(5/2))/8 + (75*(1 - 2*x)^(7/2))/56

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) (3+5 x)^2}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {847}{8 \sqrt {1-2 x}}-\frac {1133}{8} \sqrt {1-2 x}+\frac {505}{8} (1-2 x)^{3/2}-\frac {75}{8} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac {847}{8} \sqrt {1-2 x}+\frac {1133}{24} (1-2 x)^{3/2}-\frac {101}{8} (1-2 x)^{5/2}+\frac {75}{56} (1-2 x)^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{21} \sqrt {1-2 x} \left (225 x^3+723 x^2+1091 x+1469\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(Sqrt[1 - 2*x]*(1469 + 1091*x + 723*x^2 + 225*x^3))

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IntegrateAlgebraic [A]  time = 0.02, size = 40, normalized size = 0.75 \begin {gather*} \frac {1}{168} \left (225 (1-2 x)^3-2121 (1-2 x)^2+7931 (1-2 x)-17787\right ) \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-17787 + 7931*(1 - 2*x) - 2121*(1 - 2*x)^2 + 225*(1 - 2*x)^3)*Sqrt[1 - 2*x])/168

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fricas [A]  time = 0.84, size = 24, normalized size = 0.45 \begin {gather*} -\frac {1}{21} \, {\left (225 \, x^{3} + 723 \, x^{2} + 1091 \, x + 1469\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(225*x^3 + 723*x^2 + 1091*x + 1469)*sqrt(-2*x + 1)

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giac [A]  time = 1.08, size = 51, normalized size = 0.96 \begin {gather*} -\frac {75}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {101}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1133}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {847}{8} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-75/56*(2*x - 1)^3*sqrt(-2*x + 1) - 101/8*(2*x - 1)^2*sqrt(-2*x + 1) + 1133/24*(-2*x + 1)^(3/2) - 847/8*sqrt(-
2*x + 1)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {\left (225 x^{3}+723 x^{2}+1091 x +1469\right ) \sqrt {-2 x +1}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(225*x^3+723*x^2+1091*x+1469)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.58, size = 37, normalized size = 0.70 \begin {gather*} \frac {75}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {101}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1133}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {847}{8} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/56*(-2*x + 1)^(7/2) - 101/8*(-2*x + 1)^(5/2) + 1133/24*(-2*x + 1)^(3/2) - 847/8*sqrt(-2*x + 1)

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mupad [B]  time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {1133\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {847\,\sqrt {1-2\,x}}{8}-\frac {101\,{\left (1-2\,x\right )}^{5/2}}{8}+\frac {75\,{\left (1-2\,x\right )}^{7/2}}{56} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1133*(1 - 2*x)^(3/2))/24 - (847*(1 - 2*x)^(1/2))/8 - (101*(1 - 2*x)^(5/2))/8 + (75*(1 - 2*x)^(7/2))/56

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sympy [A]  time = 29.08, size = 46, normalized size = 0.87 \begin {gather*} \frac {75 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {101 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} + \frac {1133 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {847 \sqrt {1 - 2 x}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75*(1 - 2*x)**(7/2)/56 - 101*(1 - 2*x)**(5/2)/8 + 1133*(1 - 2*x)**(3/2)/24 - 847*sqrt(1 - 2*x)/8

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