3.2815 \(\int \frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{1}{9} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1877}{630} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{62092 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2835}-\frac{62092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14175}-\frac{8256877 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{56700} \]

[Out]

(-62092*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2835 - (1877*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/630 - (3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2
))/7 - (Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/9 - (8256877*Sqrt[11/3]*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/56700 - (62092*Sqrt[11/3]*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/14175

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Rubi [A]  time = 0.406342, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{1}{9} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1877}{630} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{62092 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2835}-\frac{62092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14175}-\frac{8256877 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{56700} \]

Antiderivative was successfully verified.

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

[Out]

(-62092*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2835 - (1877*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/630 - (3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2
))/7 - (Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/9 - (8256877*Sqrt[11/3]*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/56700 - (62092*Sqrt[11/3]*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/14175

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Rubi in Sympy [A]  time = 40.6319, size = 172, normalized size = 0.9 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{9} - \frac{5 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{7} - \frac{1787 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{630} - \frac{62092 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{2835} - \frac{8256877 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{170100} - \frac{62092 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{42525} \]

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)**(1/2),x)

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(5/2)/9 - 5*sqrt(-2*x + 1)*(3*x + 2)
**(3/2)*(5*x + 3)**(3/2)/7 - 1787*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/
630 - 62092*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/2835 - 8256877*sqrt(33)*e
lliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/170100 - 62092*sqrt(33)*ellipt
ic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/42525

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Mathematica [A]  time = 0.320563, size = 105, normalized size = 0.55 \[ \frac{8256877 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (47250 x^3+148950 x^2+212175 x+208073\right )+831761 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{85050 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(8256877*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]
*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(208073 + 212175*x + 148950*x^2 + 47250*x^3) + 8317
61*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(85050*Sqrt[2])

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Maple [C]  time = 0.018, size = 179, normalized size = 0.9 \[{\frac{1}{5103000\,{x}^{3}+3912300\,{x}^{2}-1190700\,x-1020600}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -42525000\,{x}^{6}+4158805\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -8256877\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -166657500\,{x}^{5}-283810500\,{x}^{4}-293881950\,{x}^{3}-72202620\,{x}^{2}[phone]\,x+37453140 \right ) } \]

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)^(1/2),x)

[Out]

1/170100*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-42525000*x^6+4158805*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))-8256877*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^
(1/2)*(1-2*x)^(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))-166657500*x^5-283810500*x^4-293881950*x^3-72202620*x^2+8188683
0*x+37453140)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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