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

Optimal. Leaf size=79 \[ \frac {1125}{352} (1-2 x)^{11/2}-\frac {4225}{96} (1-2 x)^{9/2}+\frac {28555}{112} (1-2 x)^{7/2}-\frac {64317}{80} (1-2 x)^{5/2}+\frac {48279}{32} (1-2 x)^{3/2}-\frac {65219}{32} \sqrt {1-2 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 79, 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 {1125}{352} (1-2 x)^{11/2}-\frac {4225}{96} (1-2 x)^{9/2}+\frac {28555}{112} (1-2 x)^{7/2}-\frac {64317}{80} (1-2 x)^{5/2}+\frac {48279}{32} (1-2 x)^{3/2}-\frac {65219}{32} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-65219*Sqrt[1 - 2*x])/32 + (48279*(1 - 2*x)^(3/2))/32 - (64317*(1 - 2*x)^(5/2))/80 + (28555*(1 - 2*x)^(7/2))/
112 - (4225*(1 - 2*x)^(9/2))/96 + (1125*(1 - 2*x)^(11/2))/352

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)^3}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {65219}{32 \sqrt {1-2 x}}-\frac {144837}{32} \sqrt {1-2 x}+\frac {64317}{16} (1-2 x)^{3/2}-\frac {28555}{16} (1-2 x)^{5/2}+\frac {12675}{32} (1-2 x)^{7/2}-\frac {1125}{32} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {65219}{32} \sqrt {1-2 x}+\frac {48279}{32} (1-2 x)^{3/2}-\frac {64317}{80} (1-2 x)^{5/2}+\frac {28555}{112} (1-2 x)^{7/2}-\frac {4225}{96} (1-2 x)^{9/2}+\frac {1125}{352} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {1-2 x} \left (118125 x^5+518000 x^4+1024475 x^3+1252938 x^2+1167932 x+1292672\right )}{1155} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1155*(Sqrt[1 - 2*x]*(1292672 + 1167932*x + 1252938*x^2 + 1024475*x^3 + 518000*x^4 + 118125*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 71, normalized size = 0.90 \begin {gather*} \frac {118125 (1-2 x)^{11/2}-1626625 (1-2 x)^{9/2}+9423150 (1-2 x)^{7/2}-29714454 (1-2 x)^{5/2}[phone] (1-2 x)^{3/2}-75327945 \sqrt {1-2 x}}{36960} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-75327945*Sqrt[1 - 2*x] + 55762245*(1 - 2*x)^(3/2) - 29714454*(1 - 2*x)^(5/2) + 9423150*(1 - 2*x)^(7/2) - 162
6625*(1 - 2*x)^(9/2) + 118125*(1 - 2*x)^(11/2))/36960

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fricas [A]  time = 1.47, size = 34, normalized size = 0.43 \begin {gather*} -\frac {1}{1155} \, {\left (118125 \, x^{5} + 518000 \, x^{4} + 1024475 \, x^{3} + 1252938 \, x^{2} + 1167932 \, x + 1292672\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)^3/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1155*(118125*x^5 + 518000*x^4 + 1024475*x^3 + 1252938*x^2 + 1167932*x + 1292672)*sqrt(-2*x + 1)

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giac [A]  time = 0.94, size = 83, normalized size = 1.05 \begin {gather*} -\frac {1125}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {4225}{96} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {28555}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {64317}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {48279}{32} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {65219}{32} \, \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)^3/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1125/352*(2*x - 1)^5*sqrt(-2*x + 1) - 4225/96*(2*x - 1)^4*sqrt(-2*x + 1) - 28555/112*(2*x - 1)^3*sqrt(-2*x +
1) - 64317/80*(2*x - 1)^2*sqrt(-2*x + 1) + 48279/32*(-2*x + 1)^(3/2) - 65219/32*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (118125 x^{5}+518000 x^{4}+1024475 x^{3}+1252938 x^{2}+1167932 x +1292672\right ) \sqrt {-2 x +1}}{1155} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(118125*x^5+518000*x^4+1024475*x^3+1252938*x^2+1167932*x+1292672)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.49, size = 55, normalized size = 0.70 \begin {gather*} \frac {1125}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {4225}{96} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {28555}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {64317}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {48279}{32} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {65219}{32} \, \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)^3/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

1125/352*(-2*x + 1)^(11/2) - 4225/96*(-2*x + 1)^(9/2) + 28555/112*(-2*x + 1)^(7/2) - 64317/80*(-2*x + 1)^(5/2)
+ 48279/32*(-2*x + 1)^(3/2) - 65219/32*sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {48279\,{\left (1-2\,x\right )}^{3/2}}{32}-\frac {65219\,\sqrt {1-2\,x}}{32}-\frac {64317\,{\left (1-2\,x\right )}^{5/2}}{80}+\frac {28555\,{\left (1-2\,x\right )}^{7/2}}{112}-\frac {4225\,{\left (1-2\,x\right )}^{9/2}}{96}+\frac {1125\,{\left (1-2\,x\right )}^{11/2}}{352} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(48279*(1 - 2*x)^(3/2))/32 - (65219*(1 - 2*x)^(1/2))/32 - (64317*(1 - 2*x)^(5/2))/80 + (28555*(1 - 2*x)^(7/2))
/112 - (4225*(1 - 2*x)^(9/2))/96 + (1125*(1 - 2*x)^(11/2))/352

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sympy [A]  time = 59.47, size = 70, normalized size = 0.89 \begin {gather*} \frac {1125 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} - \frac {4225 \left (1 - 2 x\right )^{\frac {9}{2}}}{96} + \frac {28555 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} - \frac {64317 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} + \frac {48279 \left (1 - 2 x\right )^{\frac {3}{2}}}{32} - \frac {65219 \sqrt {1 - 2 x}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125*(1 - 2*x)**(11/2)/352 - 4225*(1 - 2*x)**(9/2)/96 + 28555*(1 - 2*x)**(7/2)/112 - 64317*(1 - 2*x)**(5/2)/80
+ 48279*(1 - 2*x)**(3/2)/32 - 65219*sqrt(1 - 2*x)/32

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