3.1099 \(\int \frac{(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=180 \[ -\frac{a^{3/2} e^2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-9 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{a e^3 \sqrt{e x} \sqrt [4]{a+b x^2} (10 b c-9 a d)}{12 b^3}+\frac{e (e x)^{5/2} \sqrt [4]{a+b x^2} (10 b c-9 a d)}{30 b^2}+\frac{d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e} \]

[Out]

-(a*(10*b*c - 9*a*d)*e^3*Sqrt[e*x]*(a + b*x^2)^(1/4))/(12*b^3) + ((10*b*c - 9*a*
d)*e*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(30*b^2) + (d*(e*x)^(9/2)*(a + b*x^2)^(1/4))
/(5*b*e) - (a^(3/2)*(10*b*c - 9*a*d)*e^2*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Ellip
ticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(12*b^(5/2)*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.40864, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{a^{3/2} e^2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-9 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{a e^3 \sqrt{e x} \sqrt [4]{a+b x^2} (10 b c-9 a d)}{12 b^3}+\frac{e (e x)^{5/2} \sqrt [4]{a+b x^2} (10 b c-9 a d)}{30 b^2}+\frac{d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]

[Out]

-(a*(10*b*c - 9*a*d)*e^3*Sqrt[e*x]*(a + b*x^2)^(1/4))/(12*b^3) + ((10*b*c - 9*a*
d)*e*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(30*b^2) + (d*(e*x)^(9/2)*(a + b*x^2)^(1/4))
/(5*b*e) - (a^(3/2)*(10*b*c - 9*a*d)*e^2*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*Ellip
ticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(12*b^(5/2)*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 40.8395, size = 165, normalized size = 0.92 \[ \frac{a^{\frac{3}{2}} e^{2} \left (e x\right )^{\frac{3}{2}} \left (9 a d - 10 b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{12 b^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{a e^{3} \sqrt{e x} \sqrt [4]{a + b x^{2}} \left (9 a d - 10 b c\right )}{12 b^{3}} + \frac{d \left (e x\right )^{\frac{9}{2}} \sqrt [4]{a + b x^{2}}}{5 b e} - \frac{e \left (e x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}} \left (9 a d - 10 b c\right )}{30 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(d*x**2+c)/(b*x**2+a)**(3/4),x)

[Out]

a**(3/2)*e**2*(e*x)**(3/2)*(9*a*d - 10*b*c)*(a/(b*x**2) + 1)**(3/4)*elliptic_f(a
tan(sqrt(a)/(sqrt(b)*x))/2, 2)/(12*b**(5/2)*(a + b*x**2)**(3/4)) + a*e**3*sqrt(e
*x)*(a + b*x**2)**(1/4)*(9*a*d - 10*b*c)/(12*b**3) + d*(e*x)**(9/2)*(a + b*x**2)
**(1/4)/(5*b*e) - e*(e*x)**(5/2)*(a + b*x**2)**(1/4)*(9*a*d - 10*b*c)/(30*b**2)

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Mathematica [C]  time = 0.156891, size = 123, normalized size = 0.68 \[ \frac{e^3 \sqrt{e x} \left (\left (a+b x^2\right ) \left (45 a^2 d-2 a b \left (25 c+9 d x^2\right )+4 b^2 x^2 \left (5 c+3 d x^2\right )\right )+5 a^2 \left (\frac{b x^2}{a}+1\right )^{3/4} (10 b c-9 a d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )\right )}{60 b^3 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]

[Out]

(e^3*Sqrt[e*x]*((a + b*x^2)*(45*a^2*d + 4*b^2*x^2*(5*c + 3*d*x^2) - 2*a*b*(25*c
+ 9*d*x^2)) + 5*a^2*(10*b*c - 9*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4
, 3/4, 5/4, -((b*x^2)/a)]))/(60*b^3*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{7}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(3/4),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(3/4),x, algorithm="fricas")

[Out]

integral((d*e^3*x^5 + c*e^3*x^3)*sqrt(e*x)/(b*x^2 + a)^(3/4), 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((e*x)**(7/2)*(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. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(3/4),x, algorithm="giac")

[Out]

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