3.1103 \(\int \frac{c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{4 b^{3/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (6 b c-7 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac{2 \sqrt [4]{a+b x^2} (6 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}} \]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(7*a*e*(e*x)^(7/2)) + (2*(6*b*c - 7*a*d)*(a + b*x^2)^(1
/4))/(21*a^2*e^3*(e*x)^(3/2)) - (4*b^(3/2)*(6*b*c - 7*a*d)*(1 + a/(b*x^2))^(3/4)
*(e*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(21*a^(5/2)*e^6*(a + b
*x^2)^(3/4))

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Rubi [A]  time = 0.337427, antiderivative size = 144, 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 \[ -\frac{4 b^{3/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (6 b c-7 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac{2 \sqrt [4]{a+b x^2} (6 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(a + b*x^2)^(1/4))/(7*a*e*(e*x)^(7/2)) + (2*(6*b*c - 7*a*d)*(a + b*x^2)^(1
/4))/(21*a^2*e^3*(e*x)^(3/2)) - (4*b^(3/2)*(6*b*c - 7*a*d)*(1 + a/(b*x^2))^(3/4)
*(e*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(21*a^(5/2)*e^6*(a + b
*x^2)^(3/4))

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Rubi in Sympy [A]  time = 35.9221, size = 134, normalized size = 0.93 \[ - \frac{2 c \sqrt [4]{a + b x^{2}}}{7 a e \left (e x\right )^{\frac{7}{2}}} - \frac{2 \sqrt [4]{a + b x^{2}} \left (7 a d - 6 b c\right )}{21 a^{2} e^{3} \left (e x\right )^{\frac{3}{2}}} + \frac{4 b^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}} \left (7 a d - 6 b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{21 a^{\frac{5}{2}} e^{6} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*(a + b*x**2)**(1/4)/(7*a*e*(e*x)**(7/2)) - 2*(a + b*x**2)**(1/4)*(7*a*d - 6
*b*c)/(21*a**2*e**3*(e*x)**(3/2)) + 4*b**(3/2)*(e*x)**(3/2)*(7*a*d - 6*b*c)*(a/(
b*x**2) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x))/2, 2)/(21*a**(5/2)*e**6
*(a + b*x**2)**(3/4))

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Mathematica [C]  time = 0.185749, size = 107, normalized size = 0.74 \[ -\frac{2 \sqrt{e x} \left (2 b x^4 \left (\frac{b x^2}{a}+1\right )^{3/4} (7 a d-6 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+\left (a+b x^2\right ) \left (3 a c+7 a d x^2-6 b c x^2\right )\right )}{21 a^2 e^5 x^4 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[e*x]*((a + b*x^2)*(3*a*c - 6*b*c*x^2 + 7*a*d*x^2) + 2*b*(-6*b*c + 7*a*d
)*x^4*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(21
*a^2*e^5*x^4*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{9}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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