3.2645 \(\int \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=249 \[ \frac{2}{65} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{2014 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{53625}-\frac{564731 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{2252250}-\frac{1865989 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1126125}-\frac{493825477 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{40540500}-\frac{493825477 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}}-\frac{16416987253 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}} \]

[Out]

(-493825477*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/40540500 - (1865989*Sqrt[
1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/1126125 - (564731*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(5/2))/2252250 - (2014*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/
2))/53625 - (23*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/3575 + (2*Sqrt[1
- 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/65 - (16416987253*EllipticE[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/(18427500*Sqrt[33]) - (493825477*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18427500*Sqrt[33])

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Rubi [A]  time = 0.564653, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{65} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{2014 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{53625}-\frac{564731 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{2252250}-\frac{1865989 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1126125}-\frac{493825477 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{40540500}-\frac{493825477 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}}-\frac{16416987253 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-493825477*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/40540500 - (1865989*Sqrt[
1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/1126125 - (564731*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(5/2))/2252250 - (2014*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/
2))/53625 - (23*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/3575 + (2*Sqrt[1
- 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/65 - (16416987253*EllipticE[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/(18427500*Sqrt[33]) - (493825477*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18427500*Sqrt[33])

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Rubi in Sympy [A]  time = 54.5741, size = 230, normalized size = 0.92 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{39} - \frac{5 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{143} - \frac{362 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3861} - \frac{101861 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{162162} - \frac{5075047 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2027025} - \frac{472506679 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{40540500} - \frac{16416987253 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{608107500} - \frac{493825477 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{608107500} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(7/2)*(5*x + 3)**(5/2)/39 - 5*sqrt(-2*x + 1)*(3*x +
2)**(7/2)*(5*x + 3)**(3/2)/143 - 362*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**
(3/2)/3861 - 101861*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/162162 - 50750
47*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/2027025 - 472506679*sqrt(-2*x +
1)*sqrt(3*x + 2)*sqrt(5*x + 3)/40540500 - 16416987253*sqrt(33)*elliptic_e(asin(
sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/608107500 - 493825477*sqrt(33)*elliptic_f(asi
n(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/608107500

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Mathematica [A]  time = 0.445325, size = 112, normalized size = 0.45 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1403325000 x^5+4299277500 x^4+5075689500 x^3+2626854750 x^2+139824180 x-707313559\right )-16537733765 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{608107500 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-707313559 + 139824180*x + 262685
4750*x^2 + 5075689500*x^3 + 4299277500*x^4 + 1403325000*x^5) + 32833974506*Ellip
ticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 16537733765*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2])/(608107500*Sqrt[2])

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Maple [C]  time = 0.033, size = 189, normalized size = 0.8 \[{\frac{1}{36486450000\,{x}^{3}[phone]\,{x}^{2}-8513505000\,x-7297290000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1262992500000\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -32833974506\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-2005367126400\,{x}^{3}-990243288510\,{x}^{2}[phone]\,x+127316440620 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1216215000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1262992500000*x^8+483764
4000000*x^7+7239923775000*x^6+16537733765*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))-32833974506*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))[phone]
000*x^5+98606794500*x^4-2005367126400*x^3-990243288510*x^2+123367494990*x+127316
440620)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*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)**(5/2)*(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. \[ \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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