3.2753 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 (3 x+2)^{3/2}}-\frac{316 (5 x+3)^{3/2} \sqrt{1-2 x}}{27 \sqrt{3 x+2}}+\frac{5264}{243} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{5264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215}-\frac{19174 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215} \]

[Out]

(5264*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/243 - (2*(1 - 2*x)^(5/2)*(3 + 5
*x)^(3/2))/(15*(2 + 3*x)^(5/2)) + (46*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(27*(2 +
3*x)^(3/2)) - (316*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(27*Sqrt[2 + 3*x]) - (19174*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215 + (5264*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215

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Rubi [A]  time = 0.409331, antiderivative size = 191, 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 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 (3 x+2)^{3/2}}-\frac{316 (5 x+3)^{3/2} \sqrt{1-2 x}}{27 \sqrt{3 x+2}}+\frac{5264}{243} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{5264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215}-\frac{19174 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215} \]

Antiderivative was successfully verified.

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

[Out]

(5264*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/243 - (2*(1 - 2*x)^(5/2)*(3 + 5
*x)^(3/2))/(15*(2 + 3*x)^(5/2)) + (46*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(27*(2 +
3*x)^(3/2)) - (316*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(27*Sqrt[2 + 3*x]) - (19174*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215 + (5264*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215

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Rubi in Sympy [A]  time = 39.0883, size = 172, normalized size = 0.9 \[ - \frac{46 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{189 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{274 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{63 \sqrt{3 x + 2}} + \frac{5564 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1701} - \frac{19174 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3645} + \frac{8272 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{6075} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-46*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(189*(3*x + 2)**(3/2)) - 2*(-2*x + 1)**(5/2)
*(5*x + 3)**(3/2)/(15*(3*x + 2)**(5/2)) + 274*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(6
3*sqrt(3*x + 2)) + 5564*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1701 - 19174*
sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3645 + 8272*sqrt(35)
*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/6075

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Mathematica [A]  time = 0.253433, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2700 x^3+68913 x^2+83412 x+25927\right )}{(3 x+2)^{5/2}}+\sqrt{2} \left (9587 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-53015 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{3645} \]

Antiderivative was successfully verified.

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

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(25927 + 83412*x + 68913*x^2 + 2700*x^3))/(2
+ 3*x)^(5/2) + Sqrt[2]*(9587*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
- 53015*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/3645

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Maple [C]  time = 0.028, size = 391, normalized size = 2.1 \[{\frac{2}{36450\,{x}^{2}+3645\,x-10935} \left ( 477135\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-86283\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+636180\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-115044\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+212060\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -38348\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +81000\,{x}^{5}+2075490\,{x}^{4}+2684799\,{x}^{3}+407829\,{x}^{2}-672927\,x-233343 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3645*(477135*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)-86283*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)+636180*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)-115044*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)+2
12060*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))-38348*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))+81000*x^5+2075490*x^4+2684799*x^3+407829*x^2-672927*
x-233343)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((27*x^3 + 54*x
^2 + 36*x + 8)*sqrt(3*x + 2)), 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((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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