3.1942 \(\int (1-2 x)^{5/2} (2+3 x)^4 (3+5 x)^2 \, dx\)

Optimal. Leaf size=92 \[ -\frac{2025 (1-2 x)^{19/2}}{1216}+\frac{13905}{544} (1-2 x)^{17/2}-\frac{53037}{320} (1-2 x)^{15/2}+\frac{121359}{208} (1-2 x)^{13/2}-\frac{832951}{704} (1-2 x)^{11/2}+\frac{381073}{288} (1-2 x)^{9/2}-\frac{41503}{64} (1-2 x)^{7/2} \]

[Out]

(-41503*(1 - 2*x)^(7/2))/64 + (381073*(1 - 2*x)^(9/2))/288 - (832951*(1 - 2*x)^(11/2))/704 + (121359*(1 - 2*x)
^(13/2))/208 - (53037*(1 - 2*x)^(15/2))/320 + (13905*(1 - 2*x)^(17/2))/544 - (2025*(1 - 2*x)^(19/2))/1216

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Rubi [A]  time = 0.0166721, antiderivative size = 92, normalized size of antiderivative = 1., 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} \[ -\frac{2025 (1-2 x)^{19/2}}{1216}+\frac{13905}{544} (1-2 x)^{17/2}-\frac{53037}{320} (1-2 x)^{15/2}+\frac{121359}{208} (1-2 x)^{13/2}-\frac{832951}{704} (1-2 x)^{11/2}+\frac{381073}{288} (1-2 x)^{9/2}-\frac{41503}{64} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-41503*(1 - 2*x)^(7/2))/64 + (381073*(1 - 2*x)^(9/2))/288 - (832951*(1 - 2*x)^(11/2))/704 + (121359*(1 - 2*x)
^(13/2))/208 - (53037*(1 - 2*x)^(15/2))/320 + (13905*(1 - 2*x)^(17/2))/544 - (2025*(1 - 2*x)^(19/2))/1216

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 (1-2 x)^{5/2} (2+3 x)^4 (3+5 x)^2 \, dx &=\int \left (\frac{290521}{64} (1-2 x)^{5/2}-\frac{381073}{32} (1-2 x)^{7/2}+\frac{832951}{64} (1-2 x)^{9/2}-\frac{121359}{16} (1-2 x)^{11/2}+\frac{159111}{64} (1-2 x)^{13/2}-\frac{13905}{32} (1-2 x)^{15/2}+\frac{2025}{64} (1-2 x)^{17/2}\right ) \, dx\\ &=-\frac{41503}{64} (1-2 x)^{7/2}+\frac{381073}{288} (1-2 x)^{9/2}-\frac{832951}{704} (1-2 x)^{11/2}+\frac{121359}{208} (1-2 x)^{13/2}-\frac{53037}{320} (1-2 x)^{15/2}+\frac{13905}{544} (1-2 x)^{17/2}-\frac{2025 (1-2 x)^{19/2}}{1216}\\ \end{align*}

Mathematica [A]  time = 0.0215584, size = 43, normalized size = 0.47 \[ -\frac{(1-2 x)^{7/2} \left (221524875 x^6+1035520200 x^5+2092364703 x^4+2374399764 x^3+1634664492 x^2+673648856 x+138993368\right )}{2078505} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(138993368 + 673648856*x + 1634664492*x^2 + 2374399764*x^3 + 2092364703*x^4 + 1035520200*x^5
+ 221524875*x^6))/2078505

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{221524875\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+138993368}{2078505} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2078505*(221524875*x^6+1035520200*x^5+2092364703*x^4+2374399764*x^3+1634664492*x^2+673648856*x+138993368)*(
1-2*x)^(7/2)

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Maxima [A]  time = 1.34105, size = 86, normalized size = 0.93 \begin{align*} -\frac{2025}{1216} \,{\left (-2 \, x + 1\right )}^{\frac{19}{2}} + \frac{13905}{544} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} - \frac{53037}{320} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{121359}{208} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{832951}{704} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{381073}{288} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{41503}{64} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4*(3+5*x)^2,x, algorithm="maxima")

[Out]

-2025/1216*(-2*x + 1)^(19/2) + 13905/544*(-2*x + 1)^(17/2) - 53037/320*(-2*x + 1)^(15/2) + 121359/208*(-2*x +
1)^(13/2) - 832951/704*(-2*x + 1)^(11/2) + 381073/288*(-2*x + 1)^(9/2) - 41503/64*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.52219, size = 250, normalized size = 2.72 \begin{align*} \frac{1}{2078505} \,{\left (1772199000 \, x^{9} + 5625863100 \, x^{8} + 5641824474 \, x^{7} - 121581999 \, x^{6} - 3896813214 \, x^{5} - 2072749175 \, x^{4} + 461747860 \, x^{3} + 739308228 \, x^{2} + 160311352 \, x - 138993368\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4*(3+5*x)^2,x, algorithm="fricas")

[Out]

1/2078505*(1772199000*x^9 + 5625863100*x^8 + 5641824474*x^7 - 121581999*x^6 - 3896813214*x^5 - 2072749175*x^4
+ 461747860*x^3 + 739308228*x^2 + 160311352*x - 138993368)*sqrt(-2*x + 1)

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Sympy [A]  time = 20.0847, size = 82, normalized size = 0.89 \begin{align*} - \frac{2025 \left (1 - 2 x\right )^{\frac{19}{2}}}{1216} + \frac{13905 \left (1 - 2 x\right )^{\frac{17}{2}}}{544} - \frac{53037 \left (1 - 2 x\right )^{\frac{15}{2}}}{320} + \frac{121359 \left (1 - 2 x\right )^{\frac{13}{2}}}{208} - \frac{832951 \left (1 - 2 x\right )^{\frac{11}{2}}}{704} + \frac{381073 \left (1 - 2 x\right )^{\frac{9}{2}}}{288} - \frac{41503 \left (1 - 2 x\right )^{\frac{7}{2}}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2025*(1 - 2*x)**(19/2)/1216 + 13905*(1 - 2*x)**(17/2)/544 - 53037*(1 - 2*x)**(15/2)/320 + 121359*(1 - 2*x)**(
13/2)/208 - 832951*(1 - 2*x)**(11/2)/704 + 381073*(1 - 2*x)**(9/2)/288 - 41503*(1 - 2*x)**(7/2)/64

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Giac [A]  time = 2.38645, size = 153, normalized size = 1.66 \begin{align*} \frac{2025}{1216} \,{\left (2 \, x - 1\right )}^{9} \sqrt{-2 \, x + 1} + \frac{13905}{544} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} + \frac{53037}{320} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{121359}{208} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{832951}{704} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{381073}{288} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{41503}{64} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4*(3+5*x)^2,x, algorithm="giac")

[Out]

2025/1216*(2*x - 1)^9*sqrt(-2*x + 1) + 13905/544*(2*x - 1)^8*sqrt(-2*x + 1) + 53037/320*(2*x - 1)^7*sqrt(-2*x
+ 1) + 121359/208*(2*x - 1)^6*sqrt(-2*x + 1) + 832951/704*(2*x - 1)^5*sqrt(-2*x + 1) + 381073/288*(2*x - 1)^4*
sqrt(-2*x + 1) + 41503/64*(2*x - 1)^3*sqrt(-2*x + 1)