3.1054 \(\int \frac{(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx\)

Optimal. Leaf size=256 \[ \frac{6907 \sqrt{3 x^2+5 x+2}}{10010 \sqrt{x}}-\frac{6907 \sqrt{x} (3 x+2)}{10010 \sqrt{3 x^2+5 x+2}}-\frac{3693 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2002 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{6907 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5005 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}+\frac{(3445 x+1834) \sqrt{3 x^2+5 x+2}}{1001 x^{9/2}}-\frac{1231 \sqrt{3 x^2+5 x+2}}{2002 x^{3/2}}+\frac{204 \sqrt{3 x^2+5 x+2}}{385 x^{5/2}} \]

[Out]

(-6907*Sqrt[x]*(2 + 3*x))/(10010*Sqrt[2 + 5*x + 3*x^2]) + (204*Sqrt[2 + 5*x + 3*
x^2])/(385*x^(5/2)) - (1231*Sqrt[2 + 5*x + 3*x^2])/(2002*x^(3/2)) + (6907*Sqrt[2
+ 5*x + 3*x^2])/(10010*Sqrt[x]) + ((1834 + 3445*x)*Sqrt[2 + 5*x + 3*x^2])/(1001
*x^(9/2)) - (4*(11 - 25*x)*(2 + 5*x + 3*x^2)^(3/2))/(143*x^(13/2)) + (6907*(1 +
x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(5005*Sqrt[2]*Sqrt[
2 + 5*x + 3*x^2]) - (3693*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[
x]], -1/2])/(2002*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.458394, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{6907 \sqrt{3 x^2+5 x+2}}{10010 \sqrt{x}}-\frac{6907 \sqrt{x} (3 x+2)}{10010 \sqrt{3 x^2+5 x+2}}-\frac{3693 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2002 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{6907 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5005 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}+\frac{(3445 x+1834) \sqrt{3 x^2+5 x+2}}{1001 x^{9/2}}-\frac{1231 \sqrt{3 x^2+5 x+2}}{2002 x^{3/2}}+\frac{204 \sqrt{3 x^2+5 x+2}}{385 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(15/2),x]

[Out]

(-6907*Sqrt[x]*(2 + 3*x))/(10010*Sqrt[2 + 5*x + 3*x^2]) + (204*Sqrt[2 + 5*x + 3*
x^2])/(385*x^(5/2)) - (1231*Sqrt[2 + 5*x + 3*x^2])/(2002*x^(3/2)) + (6907*Sqrt[2
+ 5*x + 3*x^2])/(10010*Sqrt[x]) + ((1834 + 3445*x)*Sqrt[2 + 5*x + 3*x^2])/(1001
*x^(9/2)) - (4*(11 - 25*x)*(2 + 5*x + 3*x^2)^(3/2))/(143*x^(13/2)) + (6907*(1 +
x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(5005*Sqrt[2]*Sqrt[
2 + 5*x + 3*x^2]) - (3693*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[
x]], -1/2])/(2002*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 49.346, size = 240, normalized size = 0.94 \[ - \frac{6907 \sqrt{x} \left (6 x + 4\right )}{20020 \sqrt{3 x^{2} + 5 x + 2}} + \frac{6907 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{40040 \sqrt{3 x^{2} + 5 x + 2}} - \frac{3693 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{16016 \sqrt{3 x^{2} + 5 x + 2}} + \frac{6907 \sqrt{3 x^{2} + 5 x + 2}}{10010 \sqrt{x}} - \frac{1231 \sqrt{3 x^{2} + 5 x + 2}}{2002 x^{\frac{3}{2}}} + \frac{204 \sqrt{3 x^{2} + 5 x + 2}}{385 x^{\frac{5}{2}}} + \frac{2 \left (\frac{10335 x}{2} + 2751\right ) \sqrt{3 x^{2} + 5 x + 2}}{3003 x^{\frac{9}{2}}} - \frac{2 \left (- 50 x + 22\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{143 x^{\frac{13}{2}}} \]

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**(15/2),x)

[Out]

-6907*sqrt(x)*(6*x + 4)/(20020*sqrt(3*x**2 + 5*x + 2)) + 6907*sqrt((6*x + 4)/(x
+ 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(40040*sqrt(3*x**2 + 5*x + 2)) -
3693*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(16016*s
qrt(3*x**2 + 5*x + 2)) + 6907*sqrt(3*x**2 + 5*x + 2)/(10010*sqrt(x)) - 1231*sqrt
(3*x**2 + 5*x + 2)/(2002*x**(3/2)) + 204*sqrt(3*x**2 + 5*x + 2)/(385*x**(5/2)) +
2*(10335*x/2 + 2751)*sqrt(3*x**2 + 5*x + 2)/(3003*x**(9/2)) - 2*(-50*x + 22)*(3
*x**2 + 5*x + 2)**(3/2)/(143*x**(13/2))

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Mathematica [C]  time = 0.263755, size = 170, normalized size = 0.66 \[ \frac{-4651 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{15/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-13814 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{15/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-36930 x^7-29726 x^6+361120 x^5+840316 x^4+654400 x^3+125440 x^2-67200 x-24640}{20020 x^{13/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(15/2),x]

[Out]

(-24640 - 67200*x + 125440*x^2 + 654400*x^3 + 840316*x^4 + 361120*x^5 - 29726*x^
6 - 36930*x^7 - (13814*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(15/2)*Ellipt
icE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (4651*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[
3 + 2/x]*x^(15/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(20020*x^(13/2)*
Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.042, size = 150, normalized size = 0.6 \[{\frac{1}{60060} \left ( 2256\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{6}-6907\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{6}+124326\,{x}^{8}+96420\,{x}^{7}-6294\,{x}^{6}+1083360\,{x}^{5}+2520948\,{x}^{4}+1963200\,{x}^{3}+376320\,{x}^{2}-201600\,x-73920 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}{x}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x)

[Out]

1/60060*(2256*(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^6-6907*(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^6+124326*x^8+96420*x^7-6294
*x^6+1083360*x^5+2520948*x^4+1963200*x^3+376320*x^2-201600*x-73920)/(3*x^2+5*x+2
)^(1/2)/x^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2), x)