3.1105 \(\int \frac{(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=171 \[ \frac{e^{3/2} (4 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}+\frac{e^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}-\frac{e \sqrt{e x} (4 b c-5 a d)}{2 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}} \]

[Out]

-((4*b*c - 5*a*d)*e*Sqrt[e*x])/(2*b^2*(a + b*x^2)^(1/4)) + (d*(e*x)^(5/2))/(2*b*
e*(a + b*x^2)^(1/4)) + ((4*b*c - 5*a*d)*e^(3/2)*ArcTan[(b^(1/4)*Sqrt[e*x])/(Sqrt
[e]*(a + b*x^2)^(1/4))])/(4*b^(9/4)) + ((4*b*c - 5*a*d)*e^(3/2)*ArcTanh[(b^(1/4)
*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(9/4))

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Rubi [A]  time = 0.304934, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{e^{3/2} (4 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}+\frac{e^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}-\frac{e \sqrt{e x} (4 b c-5 a d)}{2 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]

[Out]

-((4*b*c - 5*a*d)*e*Sqrt[e*x])/(2*b^2*(a + b*x^2)^(1/4)) + (d*(e*x)^(5/2))/(2*b*
e*(a + b*x^2)^(1/4)) + ((4*b*c - 5*a*d)*e^(3/2)*ArcTan[(b^(1/4)*Sqrt[e*x])/(Sqrt
[e]*(a + b*x^2)^(1/4))])/(4*b^(9/4)) + ((4*b*c - 5*a*d)*e^(3/2)*ArcTanh[(b^(1/4)
*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(9/4))

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Rubi in Sympy [A]  time = 33.3324, size = 158, normalized size = 0.92 \[ \frac{d \left (e x\right )^{\frac{5}{2}}}{2 b e \sqrt [4]{a + b x^{2}}} + \frac{e \sqrt{e x} \left (5 a d - 4 b c\right )}{2 b^{2} \sqrt [4]{a + b x^{2}}} - \frac{e^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{9}{4}}} - \frac{e^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)*(d*x**2+c)/(b*x**2+a)**(5/4),x)

[Out]

d*(e*x)**(5/2)/(2*b*e*(a + b*x**2)**(1/4)) + e*sqrt(e*x)*(5*a*d - 4*b*c)/(2*b**2
*(a + b*x**2)**(1/4)) - e**(3/2)*(5*a*d - 4*b*c)*atan(b**(1/4)*sqrt(e*x)/(sqrt(e
)*(a + b*x**2)**(1/4)))/(4*b**(9/4)) - e**(3/2)*(5*a*d - 4*b*c)*atanh(b**(1/4)*s
qrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/(4*b**(9/4))

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Mathematica [C]  time = 0.121992, size = 84, normalized size = 0.49 \[ \frac{e \sqrt{e x} \left (\sqrt [4]{\frac{b x^2}{a}+1} (4 b c-5 a d) \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+5 a d-4 b c+b d x^2\right )}{2 b^2 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]

[Out]

(e*Sqrt[e*x]*(-4*b*c + 5*a*d + b*d*x^2 + (4*b*c - 5*a*d)*(1 + (b*x^2)/a)^(1/4)*H
ypergeometric2F1[1/4, 1/4, 5/4, -((b*x^2)/a)]))/(2*b^2*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.111, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(5/4),x)

[Out]

int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(5/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256652, size = 999, normalized size = 5.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(5/4),x, algorithm="fricas")

[Out]

1/8*(4*(b*d*e*x^2 - (4*b*c - 5*a*d)*e)*(b*x^2 + a)^(3/4)*sqrt(e*x) + 4*(b^3*x^2
+ a*b^2)*((256*b^4*c^4 - 1280*a*b^3*c^3*d + 2400*a^2*b^2*c^2*d^2 - 2000*a^3*b*c*
d^3 + 625*a^4*d^4)*e^6/b^9)^(1/4)*arctan(-(b^3*x^2 + a*b^2)*((256*b^4*c^4 - 1280
*a*b^3*c^3*d + 2400*a^2*b^2*c^2*d^2 - 2000*a^3*b*c*d^3 + 625*a^4*d^4)*e^6/b^9)^(
1/4)/((b*x^2 + a)^(3/4)*(4*b*c - 5*a*d)*sqrt(e*x)*e - (b*x^2 + a)*sqrt(((16*b^2*
c^2 - 40*a*b*c*d + 25*a^2*d^2)*sqrt(b*x^2 + a)*e^3*x + (b^5*x^2 + a*b^4)*sqrt((2
56*b^4*c^4 - 1280*a*b^3*c^3*d + 2400*a^2*b^2*c^2*d^2 - 2000*a^3*b*c*d^3 + 625*a^
4*d^4)*e^6/b^9))/(b*x^2 + a)))) + (b^3*x^2 + a*b^2)*((256*b^4*c^4 - 1280*a*b^3*c
^3*d + 2400*a^2*b^2*c^2*d^2 - 2000*a^3*b*c*d^3 + 625*a^4*d^4)*e^6/b^9)^(1/4)*log
(-((b*x^2 + a)^(3/4)*(4*b*c - 5*a*d)*sqrt(e*x)*e + (b^3*x^2 + a*b^2)*((256*b^4*c
^4 - 1280*a*b^3*c^3*d + 2400*a^2*b^2*c^2*d^2 - 2000*a^3*b*c*d^3 + 625*a^4*d^4)*e
^6/b^9)^(1/4))/(b*x^2 + a)) - (b^3*x^2 + a*b^2)*((256*b^4*c^4 - 1280*a*b^3*c^3*d
+ 2400*a^2*b^2*c^2*d^2 - 2000*a^3*b*c*d^3 + 625*a^4*d^4)*e^6/b^9)^(1/4)*log(-((
b*x^2 + a)^(3/4)*(4*b*c - 5*a*d)*sqrt(e*x)*e - (b^3*x^2 + a*b^2)*((256*b^4*c^4 -
1280*a*b^3*c^3*d + 2400*a^2*b^2*c^2*d^2 - 2000*a^3*b*c*d^3 + 625*a^4*d^4)*e^6/b
^9)^(1/4))/(b*x^2 + a)))/(b^3*x^2 + a*b^2)

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Sympy [A]  time = 162.9, size = 94, normalized size = 0.55 \[ \frac{c e^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} \Gamma \left (\frac{9}{4}\right )} + \frac{d e^{\frac{3}{2}} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(3/2)*(d*x**2+c)/(b*x**2+a)**(5/4),x)

[Out]

c*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((5/4, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/
a)/(2*a**(5/4)*gamma(9/4)) + d*e**(3/2)*x**(9/2)*gamma(9/4)*hyper((5/4, 9/4), (1
3/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/4)*gamma(13/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(5/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(5/4), x)