3.2947 \(\int \frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{137 \sqrt{3 x+2} (5 x+3)^{5/2}}{33 \sqrt{1-2 x}}-\frac{817}{66} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-91 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{91}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{12101}{20} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

-91*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (817*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]
*(3 + 5*x)^(3/2))/66 - (137*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) +
((2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (12101*Sqrt[11/3]*Ellipt
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20 - (91*Sqrt[11/3]*EllipticF[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi [A]  time = 0.404504, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{137 \sqrt{3 x+2} (5 x+3)^{5/2}}{33 \sqrt{1-2 x}}-\frac{817}{66} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-91 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{91}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{12101}{20} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-91*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (817*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]
*(3 + 5*x)^(3/2))/66 - (137*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) +
((2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (12101*Sqrt[11/3]*Ellipt
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20 - (91*Sqrt[11/3]*EllipticF[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi in Sympy [A]  time = 38.4149, size = 170, normalized size = 0.9 \[ - \frac{275 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{14} - \frac{1219 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{14} - \frac{12101 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{60} - \frac{91 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15} - \frac{137 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-275*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/14 - 1219*sqrt(-2*x + 1)*sqrt
(3*x + 2)*sqrt(5*x + 3)/14 - 12101*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x +
1)/7), 35/33)/60 - 91*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/3
3)/15 - 137*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/(21*sqrt(-2*x + 1)) + (3*x + 2)**(
3/2)*(5*x + 3)**(5/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.359931, size = 125, normalized size = 0.66 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (90 x^3+438 x^2-2579 x+957\right )-6095 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+12101 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{60 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(957 - 2579*x + 438*x^2 + 90*x^3) + 12101*Sqrt[
2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 6095*Sq
rt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(60*(
1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 286, normalized size = 1.5 \[{\frac{1}{60\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 12190\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-24202\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6095\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +12101\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -13500\,{x}^{5}-82800\,{x}^{4}+298230\,{x}^{3}+320180\,{x}^{2}-27090\,x-57420 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(12190*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)-24202*2^(1/2)*Elli
pticE(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)-6095*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))+12101*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/1
1*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-13500*x^5-82800
*x^4+298230*x^3+320180*x^2-27090*x-57420)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1
/2)/(-1+2*x)^2/(15*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((75*x^3 + 140*x^2 + 87*x + 18)*sqrt(5*x + 3)*sqrt(3*x + 2)/((4*x^2 - 4*
x + 1)*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)**(3/2)*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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