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

Optimal. Leaf size=66 \[ -\frac {45}{16} (1-2 x)^{5/2}+\frac {85}{2} (1-2 x)^{3/2}-\frac {3467}{8} \sqrt {1-2 x}-\frac {1309}{2 \sqrt {1-2 x}}+\frac {5929}{48 (1-2 x)^{3/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, 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 {45}{16} (1-2 x)^{5/2}+\frac {85}{2} (1-2 x)^{3/2}-\frac {3467}{8} \sqrt {1-2 x}-\frac {1309}{2 \sqrt {1-2 x}}+\frac {5929}{48 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

5929/(48*(1 - 2*x)^(3/2)) - 1309/(2*Sqrt[1 - 2*x]) - (3467*Sqrt[1 - 2*x])/8 + (85*(1 - 2*x)^(3/2))/2 - (45*(1
- 2*x)^(5/2))/16

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)^2 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {5929}{16 (1-2 x)^{5/2}}-\frac {1309}{2 (1-2 x)^{3/2}}+\frac {3467}{8 \sqrt {1-2 x}}-\frac {255}{2} \sqrt {1-2 x}+\frac {225}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {5929}{48 (1-2 x)^{3/2}}-\frac {1309}{2 \sqrt {1-2 x}}-\frac {3467}{8} \sqrt {1-2 x}+\frac {85}{2} (1-2 x)^{3/2}-\frac {45}{16} (1-2 x)^{5/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {135 x^4+750 x^3+3873 x^2-8430 x+2774}{3 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(2774 - 8430*x + 3873*x^2 + 750*x^3 + 135*x^4)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.03, size = 49, normalized size = 0.74 \begin {gather*} \frac {-135 (1-2 x)^4+2040 (1-2 x)^3-20802 (1-2 x)^2-31416 (1-2 x)+5929}{48 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5929 - 31416*(1 - 2*x) - 20802*(1 - 2*x)^2 + 2040*(1 - 2*x)^3 - 135*(1 - 2*x)^4)/(48*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.20, size = 41, normalized size = 0.62 \begin {gather*} -\frac {{\left (135 \, x^{4} + 750 \, x^{3} + 3873 \, x^{2} - 8430 \, x + 2774\right )} \sqrt {-2 \, x + 1}}{3 \, {\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)^2*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(135*x^4 + 750*x^3 + 3873*x^2 - 8430*x + 2774)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.24, size = 56, normalized size = 0.85 \begin {gather*} -\frac {45}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {85}{2} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3467}{8} \, \sqrt {-2 \, x + 1} - \frac {77 \, {\left (816 \, x - 331\right )}}{48 \, {\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)^2*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-45/16*(2*x - 1)^2*sqrt(-2*x + 1) + 85/2*(-2*x + 1)^(3/2) - 3467/8*sqrt(-2*x + 1) - 77/48*(816*x - 331)/((2*x
- 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \begin {gather*} -\frac {135 x^{4}+750 x^{3}+3873 x^{2}-8430 x +2774}{3 \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)^2*(5*x+3)^2/(-2*x+1)^(5/2),x)

[Out]

-1/3*(135*x^4+750*x^3+3873*x^2-8430*x+2774)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.47, size = 42, normalized size = 0.64 \begin {gather*} -\frac {45}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {85}{2} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3467}{8} \, \sqrt {-2 \, x + 1} + \frac {77 \, {\left (816 \, x - 331\right )}}{48 \, {\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)^2*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-45/16*(-2*x + 1)^(5/2) + 85/2*(-2*x + 1)^(3/2) - 3467/8*sqrt(-2*x + 1) + 77/48*(816*x - 331)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.04, size = 41, normalized size = 0.62 \begin {gather*} \frac {1309\,x-\frac {25487}{48}}{{\left (1-2\,x\right )}^{3/2}}-\frac {3467\,\sqrt {1-2\,x}}{8}+\frac {85\,{\left (1-2\,x\right )}^{3/2}}{2}-\frac {45\,{\left (1-2\,x\right )}^{5/2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1309*x - 25487/48)/(1 - 2*x)^(3/2) - (3467*(1 - 2*x)^(1/2))/8 + (85*(1 - 2*x)^(3/2))/2 - (45*(1 - 2*x)^(5/2))
/16

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sympy [A]  time = 22.47, size = 58, normalized size = 0.88 \begin {gather*} - \frac {45 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} + \frac {85 \left (1 - 2 x\right )^{\frac {3}{2}}}{2} - \frac {3467 \sqrt {1 - 2 x}}{8} - \frac {1309}{2 \sqrt {1 - 2 x}} + \frac {5929}{48 \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)**2*(3+5*x)**2/(1-2*x)**(5/2),x)

[Out]

-45*(1 - 2*x)**(5/2)/16 + 85*(1 - 2*x)**(3/2)/2 - 3467*sqrt(1 - 2*x)/8 - 1309/(2*sqrt(1 - 2*x)) + 5929/(48*(1
- 2*x)**(3/2))

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