3.2974 \(\int \frac{(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{217 (3 x+2)^{7/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3218 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 (5 x+3)^{3/2}}+\frac{110519 \sqrt{1-2 x} (3 x+2)^{3/2}}{1098075 \sqrt{5 x+3}}-\frac{5199979 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3660250}-\frac{5442127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1663750 \sqrt{33}}-\frac{90397364 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{831875 \sqrt{33}} \]

[Out]

(3218*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(19965*(3 + 5*x)^(3/2)) - (217*(2 + 3*x)^(7
/2))/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(9/2))/(33*(1 - 2*x)^(3/
2)*(3 + 5*x)^(3/2)) + (110519*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(1098075*Sqrt[3 + 5
*x]) - (5199979*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3660250 - (90397364*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(831875*Sqrt[33]) - (5442127*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1663750*Sqrt[33])

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Rubi [A]  time = 0.504048, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{217 (3 x+2)^{7/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3218 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 (5 x+3)^{3/2}}+\frac{110519 \sqrt{1-2 x} (3 x+2)^{3/2}}{1098075 \sqrt{5 x+3}}-\frac{5199979 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3660250}-\frac{5442127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1663750 \sqrt{33}}-\frac{90397364 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{831875 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(3218*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(19965*(3 + 5*x)^(3/2)) - (217*(2 + 3*x)^(7
/2))/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(9/2))/(33*(1 - 2*x)^(3/
2)*(3 + 5*x)^(3/2)) + (110519*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(1098075*Sqrt[3 + 5
*x]) - (5199979*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3660250 - (90397364*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(831875*Sqrt[33]) - (5442127*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1663750*Sqrt[33])

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Rubi in Sympy [A]  time = 45.6203, size = 201, normalized size = 0.92 \[ \frac{3218 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{19965 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{110519 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{1098075 \sqrt{5 x + 3}} - \frac{5199979 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{3660250} - \frac{90397364 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{27451875} - \frac{5442127 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{54903750} - \frac{217 \left (3 x + 2\right )^{\frac{7}{2}}}{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \left (3 x + 2\right )^{\frac{9}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(11/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

3218*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(19965*(5*x + 3)**(3/2)) + 110519*sqrt(-2*x
+ 1)*(3*x + 2)**(3/2)/(1098075*sqrt(5*x + 3)) - 5199979*sqrt(-2*x + 1)*sqrt(3*x
+ 2)*sqrt(5*x + 3)/3660250 - 90397364*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2
*x + 1)/7), 35/33)/27451875 - 5442127*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*
x + 1)/7), 35/33)/54903750 - 217*(3*x + 2)**(7/2)/(121*sqrt(-2*x + 1)*(5*x + 3)*
*(3/2)) + 7*(3*x + 2)**(9/2)/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.40428, size = 112, normalized size = 0.51 \[ \frac{-\frac{10 \sqrt{3 x+2} \left (177888150 x^4-1825153850 x^3-1696384053 x^2+89252928 x+246962693\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}-181999265 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{109807500} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

((-10*Sqrt[2 + 3*x]*(246962693 + 89252928*x - 1696384053*x^2 - 1825153850*x^3 +
177888150*x^4))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 361589456*Sqrt[2]*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 181999265*Sqrt[2]*EllipticF[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2])/109807500

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Maple [C]  time = 0.034, size = 388, normalized size = 1.8 \[{\frac{1}{109807500\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 1819992650\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3615894560\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-361589456\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-545997795\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5336644500\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-9193939350\,x-4939253860 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/109807500*(1-2*x)^(1/2)*(1819992650*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3
+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2
*x)^(1/2)-3615894560*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I
*11^(1/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+1819992
65*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*
2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-361589456*2^(1/2)*EllipticE
(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(
1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-545997795*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(
1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2
)*2^(1/2))[phone]*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elliptic
E(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-5336644500
*x^5+51196852500*x^4+87394598590*x^3+31250093220*x^2-9193939350*x-4939253860)/(3
+5*x)^(3/2)/(-1+2*x)^2/(2+3*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="fricas")

[Out]

integral((243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*sqrt(3*x + 2)/((1
00*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**(11/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)