3.1050 \(\int \frac{(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{7/2}} \, dx\)

Optimal. Leaf size=185 \[ \frac{2 (89-35 x) \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}-\frac{1418 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}-\frac{117 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{1418 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}}-\frac{4 (3-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{15 x^{5/2}} \]

[Out]

(-1418*Sqrt[x]*(2 + 3*x))/(15*Sqrt[2 + 5*x + 3*x^2]) + (2*(89 - 35*x)*Sqrt[2 + 5
*x + 3*x^2])/(5*Sqrt[x]) - (4*(3 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/(15*x^(5/2)) +
(1418*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/
(15*Sqrt[2 + 5*x + 3*x^2]) - (117*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Ellipt
icF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi [A]  time = 0.290399, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{2 (89-35 x) \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}-\frac{1418 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}-\frac{117 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{1418 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}}-\frac{4 (3-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{15 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-1418*Sqrt[x]*(2 + 3*x))/(15*Sqrt[2 + 5*x + 3*x^2]) + (2*(89 - 35*x)*Sqrt[2 + 5
*x + 3*x^2])/(5*Sqrt[x]) - (4*(3 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/(15*x^(5/2)) +
(1418*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/
(15*Sqrt[2 + 5*x + 3*x^2]) - (117*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Ellipt
icF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi in Sympy [A]  time = 32.2366, size = 175, normalized size = 0.95 \[ - \frac{709 \sqrt{x} \left (6 x + 4\right )}{15 \sqrt{3 x^{2} + 5 x + 2}} + \frac{709 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{30 \sqrt{3 x^{2} + 5 x + 2}} - \frac{117 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} + \frac{4 \left (- \frac{105 x}{2} + \frac{267}{2}\right ) \sqrt{3 x^{2} + 5 x + 2}}{15 \sqrt{x}} - \frac{2 \left (- 10 x + 6\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{15 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-709*sqrt(x)*(6*x + 4)/(15*sqrt(3*x**2 + 5*x + 2)) + 709*sqrt((6*x + 4)/(x + 1))
*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(30*sqrt(3*x**2 + 5*x + 2)) - 117*sqr
t((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(4*sqrt(3*x**2 +
5*x + 2)) + 4*(-105*x/2 + 267/2)*sqrt(3*x**2 + 5*x + 2)/(15*sqrt(x)) - 2*(-10*x
+ 6)*(3*x**2 + 5*x + 2)**(3/2)/(15*x**(5/2))

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Mathematica [C]  time = 0.266399, size = 163, normalized size = 0.88 \[ \frac{-337 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-1418 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (225 x^5+1605 x^4+2230 x^3+906 x^2+80 x+24\right )}{15 x^{5/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(24 + 80*x + 906*x^2 + 2230*x^3 + 1605*x^4 + 225*x^5) - (1418*I)*Sqrt[2]*Sqr
t[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2]
- (337*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(7/2)*EllipticF[I*ArcSinh[Sq
rt[2/3]/Sqrt[x]], 3/2])/(15*x^(5/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 135, normalized size = 0.7 \[{\frac{1}{45} \left ( 372\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-709\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-1350\,{x}^{5}+3132\,{x}^{4}+7890\,{x}^{3}+3072\,{x}^{2}-480\,x-144 \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(372*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*EllipticF(1/2*(
6*x+4)^(1/2),I*2^(1/2))*x^2-709*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)
^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))*x^2-1350*x^5+3132*x^4+7890*x^3+307
2*x^2-480*x-144)/(3*x^2+5*x+2)^(1/2)/x^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{x^{\frac{7}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(15*x^3 + 19*x^2 - 4)*sqrt(3*x^2 + 5*x + 2)/x^(7/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{4 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{7}{2}}}\right )\, dx - \int \frac{19 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{2}}}\, dx - \int \frac{15 \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-4*sqrt(3*x**2 + 5*x + 2)/x**(7/2), x) - Integral(19*sqrt(3*x**2 + 5*x
+ 2)/x**(3/2), x) - Integral(15*sqrt(3*x**2 + 5*x + 2)/sqrt(x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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