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

Optimal. Leaf size=105 \[ \frac {10125 (1-2 x)^{11/2}}{1408}-\frac {17925}{128} (1-2 x)^{9/2}+\frac {1101465}{896} (1-2 x)^{7/2}-\frac {4177401}{640} (1-2 x)^{5/2}+\frac {9504551}{384} (1-2 x)^{3/2}-\frac {12973191}{128} \sqrt {1-2 x}-\frac {9836211}{128 \sqrt {1-2 x}}+\frac {3195731}{384 (1-2 x)^{3/2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 105, 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 {10125 (1-2 x)^{11/2}}{1408}-\frac {17925}{128} (1-2 x)^{9/2}+\frac {1101465}{896} (1-2 x)^{7/2}-\frac {4177401}{640} (1-2 x)^{5/2}+\frac {9504551}{384} (1-2 x)^{3/2}-\frac {12973191}{128} \sqrt {1-2 x}-\frac {9836211}{128 \sqrt {1-2 x}}+\frac {3195731}{384 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3195731/(384*(1 - 2*x)^(3/2)) - 9836211/(128*Sqrt[1 - 2*x]) - (12973191*Sqrt[1 - 2*x])/128 + (9504551*(1 - 2*x
)^(3/2))/384 - (4177401*(1 - 2*x)^(5/2))/640 + (1101465*(1 - 2*x)^(7/2))/896 - (17925*(1 - 2*x)^(9/2))/128 + (
10125*(1 - 2*x)^(11/2))/1408

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)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {3195731}{128 (1-2 x)^{5/2}}-\frac {9836211}{128 (1-2 x)^{3/2}}+\frac {12973191}{128 \sqrt {1-2 x}}-\frac {9504551}{128} \sqrt {1-2 x}+\frac {4177401}{128} (1-2 x)^{3/2}-\frac {1101465}{128} (1-2 x)^{5/2}+\frac {161325}{128} (1-2 x)^{7/2}-\frac {10125}{128} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac {3195731}{384 (1-2 x)^{3/2}}-\frac {9836211}{128 \sqrt {1-2 x}}-\frac {12973191}{128} \sqrt {1-2 x}+\frac {9504551}{384} (1-2 x)^{3/2}-\frac {4177401}{640} (1-2 x)^{5/2}+\frac {1101465}{896} (1-2 x)^{7/2}-\frac {17925}{128} (1-2 x)^{9/2}+\frac {10125 (1-2 x)^{11/2}}{1408}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} -\frac {1063125 x^7+6630750 x^6+19961775 x^5+41201532 x^4+77493296 x^3+258342648 x^2-522173856 x+173891632}{1155 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1155*(173891632 - 522173856*x + 258342648*x^2 + 77493296*x^3 + 41201532*x^4 + 19961775*x^5 + 6630750*x^6 +
1063125*x^7)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.03, size = 76, normalized size = 0.72 \begin {gather*} \frac {1063125 (1-2 x)^7-20703375 (1-2 x)^6+181741725 (1-2 x)^5-964979631 (1-2 x)^4+3659252135 (1-2 x)^3-14984035605 (1-2 x)^2-11360823705 (1-2 x)[phone]}{147840 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1230356435 - 11360823705*(1 - 2*x) - 14984035605*(1 - 2*x)^2 + 3659252135*(1 - 2*x)^3 - 964979631*(1 - 2*x)^4
+ 181741725*(1 - 2*x)^5 - 20703375*(1 - 2*x)^6 + 1063125*(1 - 2*x)^7)/(147840*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.26, size = 56, normalized size = 0.53 \begin {gather*} -\frac {{\left (1063125 \, x^{7} + 6630750 \, x^{6} + 19961775 \, x^{5} + 41201532 \, x^{4} + 77493296 \, x^{3} + 258342648 \, x^{2} - 522173856 \, x + 173891632\right )} \sqrt {-2 \, x + 1}}{1155 \, {\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)^4*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/1155*(1063125*x^7 + 6630750*x^6 + 19961775*x^5 + 41201532*x^4 + 77493296*x^3 + 258342648*x^2 - 522173856*x
+ 173891632)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.31, size = 104, normalized size = 0.99 \begin {gather*} -\frac {10125}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {17925}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1101465}{896} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {4177401}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {9504551}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {12973191}{128} \, \sqrt {-2 \, x + 1} - \frac {41503 \, {\left (711 \, x - 317\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)^4*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-10125/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 17925/128*(2*x - 1)^4*sqrt(-2*x + 1) - 1101465/896*(2*x - 1)^3*sqrt(-
2*x + 1) - 4177401/640*(2*x - 1)^2*sqrt(-2*x + 1) + 9504551/384*(-2*x + 1)^(3/2) - 12973191/128*sqrt(-2*x + 1)
- 41503/192*(711*x - 317)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 45, normalized size = 0.43 \begin {gather*} -\frac {1063125 x^{7}+6630750 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-522173856 x [phone]}{1155 \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)^4*(5*x+3)^3/(-2*x+1)^(5/2),x)

[Out]

-1/1155*(1063125*x^7+6630750*x^6+19961775*x^5+41201532*x^4+77493296*x^3+258342648*x^2-522173856*x+173891632)/(
-2*x+1)^(3/2)

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maxima [A]  time = 0.52, size = 69, normalized size = 0.66 \begin {gather*} \frac {10125}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {17925}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1101465}{896} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {4177401}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {9504551}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {12973191}{128} \, \sqrt {-2 \, x + 1} + \frac {41503 \, {\left (711 \, x - 317\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)^4*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

10125/1408*(-2*x + 1)^(11/2) - 17925/128*(-2*x + 1)^(9/2) + 1101465/896*(-2*x + 1)^(7/2) - 4177401/640*(-2*x +
1)^(5/2) + 9504551/384*(-2*x + 1)^(3/2) - 12973191/128*sqrt(-2*x + 1) + 41503/192*(711*x - 317)/(-2*x + 1)^(3
/2)

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mupad [B]  time = 0.03, size = 68, normalized size = 0.65 \begin {gather*} \frac {\frac {9836211\,x}{64}-\frac {13156451}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {12973191\,\sqrt {1-2\,x}}{128}+\frac {9504551\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {4177401\,{\left (1-2\,x\right )}^{5/2}}{640}+\frac {1101465\,{\left (1-2\,x\right )}^{7/2}}{896}-\frac {17925\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {10125\,{\left (1-2\,x\right )}^{11/2}}{1408} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((9836211*x)/64 - 13156451/192)/(1 - 2*x)^(3/2) - (12973191*(1 - 2*x)^(1/2))/128 + (9504551*(1 - 2*x)^(3/2))/3
84 - (4177401*(1 - 2*x)^(5/2))/640 + (1101465*(1 - 2*x)^(7/2))/896 - (17925*(1 - 2*x)^(9/2))/128 + (10125*(1 -
2*x)^(11/2))/1408

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sympy [A]  time = 43.40, size = 94, normalized size = 0.90 \begin {gather*} \frac {10125 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {17925 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {1101465 \left (1 - 2 x\right )^{\frac {7}{2}}}{896} - \frac {4177401 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} + \frac {9504551 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} - \frac {12973191 \sqrt {1 - 2 x}}{128} - \frac {9836211}{128 \sqrt {1 - 2 x}} + \frac {3195731}{384 \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)**4*(3+5*x)**3/(1-2*x)**(5/2),x)

[Out]

10125*(1 - 2*x)**(11/2)/1408 - 17925*(1 - 2*x)**(9/2)/128 + 1101465*(1 - 2*x)**(7/2)/896 - 4177401*(1 - 2*x)**
(5/2)/640 + 9504551*(1 - 2*x)**(3/2)/384 - 12973191*sqrt(1 - 2*x)/128 - 9836211/(128*sqrt(1 - 2*x)) + 3195731/
(384*(1 - 2*x)**(3/2))

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