3.1093 \(\int \frac{(e x)^{5/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=173 \[ \frac{3 a e^{5/2} (8 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}-\frac{3 a e^{5/2} (8 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}+\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (8 b c-7 a d)}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e} \]

[Out]

((8*b*c - 7*a*d)*e*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(16*b^2) + (d*(e*x)^(7/2)*(a +
b*x^2)^(1/4))/(4*b*e) + (3*a*(8*b*c - 7*a*d)*e^(5/2)*ArcTan[(b^(1/4)*Sqrt[e*x])
/(Sqrt[e]*(a + b*x^2)^(1/4))])/(32*b^(11/4)) - (3*a*(8*b*c - 7*a*d)*e^(5/2)*ArcT
anh[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(32*b^(11/4))

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Rubi [A]  time = 0.360219, antiderivative size = 173, 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 \[ \frac{3 a e^{5/2} (8 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}-\frac{3 a e^{5/2} (8 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}+\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (8 b c-7 a d)}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e} \]

Antiderivative was successfully verified.

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

[Out]

((8*b*c - 7*a*d)*e*(e*x)^(3/2)*(a + b*x^2)^(1/4))/(16*b^2) + (d*(e*x)^(7/2)*(a +
b*x^2)^(1/4))/(4*b*e) + (3*a*(8*b*c - 7*a*d)*e^(5/2)*ArcTan[(b^(1/4)*Sqrt[e*x])
/(Sqrt[e]*(a + b*x^2)^(1/4))])/(32*b^(11/4)) - (3*a*(8*b*c - 7*a*d)*e^(5/2)*ArcT
anh[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(32*b^(11/4))

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Rubi in Sympy [A]  time = 35.4988, size = 165, normalized size = 0.95 \[ - \frac{3 a e^{\frac{5}{2}} \left (7 a d - 8 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{32 b^{\frac{11}{4}}} + \frac{3 a e^{\frac{5}{2}} \left (7 a d - 8 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{32 b^{\frac{11}{4}}} + \frac{d \left (e x\right )^{\frac{7}{2}} \sqrt [4]{a + b x^{2}}}{4 b e} - \frac{e \left (e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x^{2}} \left (7 a d - 8 b c\right )}{16 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a*e**(5/2)*(7*a*d - 8*b*c)*atan(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/
4)))/(32*b**(11/4)) + 3*a*e**(5/2)*(7*a*d - 8*b*c)*atanh(b**(1/4)*sqrt(e*x)/(sqr
t(e)*(a + b*x**2)**(1/4)))/(32*b**(11/4)) + d*(e*x)**(7/2)*(a + b*x**2)**(1/4)/(
4*b*e) - e*(e*x)**(3/2)*(a + b*x**2)**(1/4)*(7*a*d - 8*b*c)/(16*b**2)

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Mathematica [C]  time = 0.13174, size = 97, normalized size = 0.56 \[ \frac{e (e x)^{3/2} \left (a \left (\frac{b x^2}{a}+1\right )^{3/4} (7 a d-8 b c) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-\left (a+b x^2\right ) \left (7 a d-4 b \left (2 c+d x^2\right )\right )\right )}{16 b^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(e*(e*x)^(3/2)*(-((a + b*x^2)*(7*a*d - 4*b*(2*c + d*x^2))) + a*(-8*b*c + 7*a*d)*
(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -((b*x^2)/a)]))/(16*b^2*(
a + b*x^2)^(3/4))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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