3.1127 \(\int \frac{(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=149 \[ \frac{d e^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac{d e^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}-\frac{2 d e \sqrt{e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{5/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.247068, antiderivative size = 149, 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 \[ \frac{d e^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac{d e^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}-\frac{2 d e \sqrt{e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{5/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.3222, size = 138, normalized size = 0.93 \[ - \frac{2 d e \sqrt{e x}}{b^{2} \sqrt [4]{a + b x^{2}}} + \frac{d e^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{9}{4}}} + \frac{d e^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{9}{4}}} - \frac{2 \left (e x\right )^{\frac{5}{2}} \left (a d - b c\right )}{5 a b e \left (a + b x^{2}\right )^{\frac{5}{4}}} \]

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)**(9/4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.127835, size = 96, normalized size = 0.64 \[ \frac{2 e \sqrt{e x} \left (-5 a^2 d+5 a d \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-6 a b d x^2+b^2 c x^2\right )}{5 a b^2 \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

(2*e*Sqrt[e*x]*(-5*a^2*d + b^2*c*x^2 - 6*a*b*d*x^2 + 5*a*d*(a + b*x^2)*(1 + (b*x
^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, -((b*x^2)/a)]))/(5*a*b^2*(a + b*x^
2)^(5/4))

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

[Out]

int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(9/4), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.271478, size = 564, normalized size = 3.79 \[ -\frac{4 \,{\left (5 \, a^{2} d e -{\left (b^{2} c - 6 \, a b d\right )} e x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} + 20 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d e +{\left (b x^{2} + a\right )} \sqrt{\frac{\sqrt{b x^{2} + a} d^{2} e^{3} x +{\left (b^{5} x^{2} + a b^{4}\right )} \sqrt{\frac{d^{4} e^{6}}{b^{9}}}}{b x^{2} + a}}}\right ) - 5 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d e +{\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}}}{b x^{2} + a}\right ) + 5 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d e -{\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}}}{b x^{2} + a}\right )}{10 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(9/4),x, algorithm="fricas")

[Out]

-1/10*(4*(5*a^2*d*e - (b^2*c - 6*a*b*d)*e*x^2)*(b*x^2 + a)^(3/4)*sqrt(e*x) + 20*
(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(d^4*e^6/b^9)^(1/4)*arctan((b^3*x^2 + a*b^
2)*(d^4*e^6/b^9)^(1/4)/((b*x^2 + a)^(3/4)*sqrt(e*x)*d*e + (b*x^2 + a)*sqrt((sqrt
(b*x^2 + a)*d^2*e^3*x + (b^5*x^2 + a*b^4)*sqrt(d^4*e^6/b^9))/(b*x^2 + a)))) - 5*
(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(d^4*e^6/b^9)^(1/4)*log(((b*x^2 + a)^(3/4)
*sqrt(e*x)*d*e + (b^3*x^2 + a*b^2)*(d^4*e^6/b^9)^(1/4))/(b*x^2 + a)) + 5*(a*b^4*
x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(d^4*e^6/b^9)^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e
*x)*d*e - (b^3*x^2 + a*b^2)*(d^4*e^6/b^9)^(1/4))/(b*x^2 + a)))/(a*b^4*x^4 + 2*a^
2*b^3*x^2 + a^3*b^2)

_______________________________________________________________________________________

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)**(3/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(9/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(9/4), x)