3.1117 \(\int \frac{\sqrt{e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{d \sqrt{e} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{2 (e x)^{3/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]

[Out]

(2*(b*c - a*d)*(e*x)^(3/2))/(3*a*b*e*(a + b*x^2)^(3/4)) - (d*Sqrt[e]*ArcTan[(b^(
1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/b^(7/4) + (d*Sqrt[e]*ArcTanh[(b^(1
/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/b^(7/4)

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Rubi [A]  time = 0.243671, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{d \sqrt{e} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{2 (e x)^{3/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]

[Out]

(2*(b*c - a*d)*(e*x)^(3/2))/(3*a*b*e*(a + b*x^2)^(3/4)) - (d*Sqrt[e]*ArcTan[(b^(
1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/b^(7/4) + (d*Sqrt[e]*ArcTanh[(b^(1
/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/b^(7/4)

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Rubi in Sympy [A]  time = 29.3493, size = 112, normalized size = 0.9 \[ - \frac{d \sqrt{e} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{7}{4}}} + \frac{d \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{7}{4}}} - \frac{2 \left (e x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)

[Out]

-d*sqrt(e)*atan(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/b**(7/4) + d*s
qrt(e)*atanh(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)**(1/4)))/b**(7/4) - 2*(e*x
)**(3/2)*(a*d - b*c)/(3*a*b*e*(a + b*x**2)**(3/4))

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Mathematica [C]  time = 0.0663421, size = 73, normalized size = 0.58 \[ \frac{2 x \sqrt{e x} \left (a d \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-a d+b c\right )}{3 a b \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]

[Out]

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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c)\sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)

[Out]

int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ d \sqrt{e} \int \frac{x^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} + \frac{2 \, c \sqrt{e} x^{\frac{3}{2}}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(7/4),x, algorithm="maxima")

[Out]

d*sqrt(e)*integrate(x^(5/2)/(b*x^2 + a)^(7/4), x) + 2/3*c*sqrt(e)*x^(3/2)/((b*x^
2 + a)^(3/4)*a)

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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)*sqrt(e*x)/(b*x^2 + a)^(7/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 173.65, size = 87, normalized size = 0.7 \[ \frac{c \sqrt{e} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )}{2 a^{\frac{7}{4}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} \Gamma \left (\frac{7}{4}\right )} + \frac{d \sqrt{e} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{7}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{7}{4}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)

[Out]

c*sqrt(e)*x**(3/2)*gamma(3/4)/(2*a**(7/4)*(1 + b*x**2/a)**(3/4)*gamma(7/4)) + d*
sqrt(e)*x**(7/2)*gamma(7/4)*hyper((7/4, 7/4), (11/4,), b*x**2*exp_polar(I*pi)/a)
/(2*a**(7/4)*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(7/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(7/4), x)