3.1125 \(\int \frac{c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{8 b^{3/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-7 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{7/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac{4 \sqrt [4]{a+b x^2} (10 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}} \]

[Out]

(-2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(3/4)) - (2*(10*b*c - 7*a*d))/(21*a^2*e^3*
(e*x)^(3/2)*(a + b*x^2)^(3/4)) + (4*(10*b*c - 7*a*d)*(a + b*x^2)^(1/4))/(21*a^3*
e^3*(e*x)^(3/2)) - (8*b^(3/2)*(10*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^(7/2)*e^6*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.393772, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{8 b^{3/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-7 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{7/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac{4 \sqrt [4]{a+b x^2} (10 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(7/4)),x]

[Out]

(-2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(3/4)) - (2*(10*b*c - 7*a*d))/(21*a^2*e^3*
(e*x)^(3/2)*(a + b*x^2)^(3/4)) + (4*(10*b*c - 7*a*d)*(a + b*x^2)^(1/4))/(21*a^3*
e^3*(e*x)^(3/2)) - (8*b^(3/2)*(10*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^(7/2)*e^6*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 42.5803, size = 170, normalized size = 0.94 \[ - \frac{2 c}{7 a e \left (e x\right )^{\frac{7}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (7 a d - 10 b c\right )}{21 a^{2} e^{3} \left (e x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{4 \sqrt [4]{a + b x^{2}} \left (7 a d - 10 b c\right )}{21 a^{3} e^{3} \left (e x\right )^{\frac{3}{2}}} + \frac{8 b^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}} \left (7 a d - 10 b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{21 a^{\frac{7}{2}} e^{6} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

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)**(7/4),x)

[Out]

-2*c/(7*a*e*(e*x)**(7/2)*(a + b*x**2)**(3/4)) + 2*(7*a*d - 10*b*c)/(21*a**2*e**3
*(e*x)**(3/2)*(a + b*x**2)**(3/4)) - 4*(a + b*x**2)**(1/4)*(7*a*d - 10*b*c)/(21*
a**3*e**3*(e*x)**(3/2)) + 8*b**(3/2)*(e*x)**(3/2)*(7*a*d - 10*b*c)*(a/(b*x**2) +
1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x))/2, 2)/(21*a**(7/2)*e**6*(a + b*x
**2)**(3/4))

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Mathematica [C]  time = 0.241079, size = 121, normalized size = 0.67 \[ \frac{\sqrt{e x} \left (-2 a^2 \left (3 c+7 d x^2\right )+8 b x^4 \left (\frac{b x^2}{a}+1\right )^{3/4} (10 b c-7 a d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+4 a b x^2 \left (5 c-7 d x^2\right )+40 b^2 c x^4\right )}{21 a^3 e^5 x^4 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(7/4)),x]

[Out]

(Sqrt[e*x]*(40*b^2*c*x^4 + 4*a*b*x^2*(5*c - 7*d*x^2) - 2*a^2*(3*c + 7*d*x^2) + 8
*b*(10*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^3*e^5*x^4*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{9}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(7/4),x)

[Out]

int((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(9/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(9/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b e^{4} x^{6} + a e^{4} x^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(9/2)),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)/((b*e^4*x^6 + a*e^4*x^4)*(b*x^2 + a)^(3/4)*sqrt(e*x)), 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)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(9/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(9/2)), x)