3.2813 \(\int \frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{1}{13} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac{41}{143} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac{14303 \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac{221673 \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}}{50050}-\frac{138809831 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4504500}-\frac{2295970088 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{10135125}-\frac{2295970088 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4606875 \sqrt{33}}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}} \]

[Out]

(-2295970088*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/10135125 - (138809831*Sq
rt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/4504500 - (221673*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(5/2))/50050 - (14303*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)
^(5/2))/12870 - (41*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/143 - (Sqrt[1
- 2*x]*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/13 - (610627101631*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(36855000*Sqrt[33]) - (2295970088*EllipticF[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4606875*Sqrt[33])

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Rubi [A]  time = 0.572735, 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 \[ -\frac{1}{13} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac{41}{143} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac{14303 \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac{221673 \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}}{50050}-\frac{138809831 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4504500}-\frac{2295970088 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{10135125}-\frac{2295970088 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4606875 \sqrt{33}}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]

[Out]

(-2295970088*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/10135125 - (138809831*Sq
rt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/4504500 - (221673*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(5/2))/50050 - (14303*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)
^(5/2))/12870 - (41*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/143 - (Sqrt[1
- 2*x]*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/13 - (610627101631*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(36855000*Sqrt[33]) - (2295970088*EllipticF[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4606875*Sqrt[33])

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Rubi in Sympy [A]  time = 57.827, size = 230, normalized size = 0.92 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{13} - \frac{205 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{429} - \frac{13565 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{7722} - \frac{947468 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{81081} - \frac{377529563 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{8108100} - \frac{8787401429 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{40540500} - \frac{610627101631 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1216215000} - \frac{2295970088 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{161240625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(7/2)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**(7/2)*(5*x + 3)**(5/2)/13 - 205*sqrt(-2*x + 1)*(3*x +
2)**(7/2)*(5*x + 3)**(3/2)/429 - 13565*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3
)**(3/2)/7722 - 947468*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/81081 - 377
529563*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/8108100 - 8787401429*sqrt(-
2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/40540500 - 610627101631*sqrt(33)*elliptic_e
(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1216215000 - 2295970088*sqrt(35)*ellipt
ic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/161240625

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Mathematica [A]  time = 0.525774, size = 115, normalized size = 0.46 \[ \frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2104987500 x^5+9351247500 x^4+18620894250 x^3+22592085750 x^2+19961825445 x+16001700059\right )[phone] F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{608107500 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]

[Out]

(610627101631*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]*(16001700059 + 19961825445*x + 22592085750*x^2
+ 18620894250*x^3 + 9351247500*x^4 + 2104987500*x^5) + 61511810003*EllipticF[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(608107500*Sqrt[2])

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Maple [C]  time = 0.019, size = 189, normalized size = 0.8 \[{\frac{1}{36486450000\,{x}^{3}[phone]\,{x}^{2}-8513505000\,x-7297290000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -1894488750000\,{x}^{8}-9868564125000\,{x}^{7}-22769118225000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -610627101631\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -30838634482500\,{x}^{5}-27960569725500\,{x}^{4}-20079090637650\,{x}^{3}-2782614262260\,{x}^{2}[phone]\,x+2880306010620 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(7/2)*(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)*(-1894488750000*x^8-98685
64125000*x^7-22769118225000*x^6+307559050015*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))-610627101631*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))-308386
34482500*x^5-27960569725500*x^4-20079090637650*x^3-2782614262260*x^2+69534855924
90*x+2880306010620)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/sqrt(-2*x + 1),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/sqrt(-2*x + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{\sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/sqrt(-2*x + 1),x, algorithm="fricas")

[Out]

integral((675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*sqrt(5*x + 3)*s
qrt(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)**(7/2)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/sqrt(-2*x + 1),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/sqrt(-2*x + 1), x)