3.1138 \(\int \frac{c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=219 \[ \frac{16 b^{3/2} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (14 b c-9 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 a^{9/2} e^6 \sqrt [4]{a+b x^2}}-\frac{8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac{2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}} \]

[Out]

(-2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(5/4)) - (2*(14*b*c - 9*a*d))/(45*a^2*e^3*
(e*x)^(5/2)*(a + b*x^2)^(5/4)) + (4*(14*b*c - 9*a*d))/(45*a^3*e^3*(e*x)^(5/2)*(a
+ b*x^2)^(1/4)) - (8*b*(14*b*c - 9*a*d))/(15*a^4*e^5*Sqrt[e*x]*(a + b*x^2)^(1/4
)) + (16*b^(3/2)*(14*b*c - 9*a*d)*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*EllipticE[ArcC
ot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(15*a^(9/2)*e^6*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.385339, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{16 b^{3/2} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (14 b c-9 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 a^{9/2} e^6 \sqrt [4]{a+b x^2}}-\frac{8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac{2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(9/4)),x]

[Out]

(-2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(5/4)) - (2*(14*b*c - 9*a*d))/(45*a^2*e^3*
(e*x)^(5/2)*(a + b*x^2)^(5/4)) + (4*(14*b*c - 9*a*d))/(45*a^3*e^3*(e*x)^(5/2)*(a
+ b*x^2)^(1/4)) - (8*b*(14*b*c - 9*a*d))/(15*a^4*e^5*Sqrt[e*x]*(a + b*x^2)^(1/4
)) + (16*b^(3/2)*(14*b*c - 9*a*d)*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*EllipticE[ArcC
ot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(15*a^(9/2)*e^6*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 c}{9 a e \left (e x\right )^{\frac{9}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \left (9 a d - 14 b c\right )}{45 a^{2} e^{3} \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} - \frac{4 \left (9 a d - 14 b c\right )}{45 a^{3} e^{3} \left (e x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}}} + \frac{8 b \left (9 a d - 14 b c\right )}{15 a^{4} e^{5} \sqrt{e x} \sqrt [4]{a + b x^{2}}} + \frac{8 b \sqrt{e x} \left (9 a d - 14 b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{15 a^{4} e^{6} \sqrt [4]{a + b x^{2}}} - \frac{16 b \sqrt{e x} \left (9 a d - 14 b c\right )}{15 a^{4} e^{6} x \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(11/2)/(b*x**2+a)**(9/4),x)

[Out]

-2*c/(9*a*e*(e*x)**(9/2)*(a + b*x**2)**(5/4)) + 2*(9*a*d - 14*b*c)/(45*a**2*e**3
*(e*x)**(5/2)*(a + b*x**2)**(5/4)) - 4*(9*a*d - 14*b*c)/(45*a**3*e**3*(e*x)**(5/
2)*(a + b*x**2)**(1/4)) + 8*b*(9*a*d - 14*b*c)/(15*a**4*e**5*sqrt(e*x)*(a + b*x*
*2)**(1/4)) + 8*b*sqrt(e*x)*(9*a*d - 14*b*c)*(a/(b*x**2) + 1)**(1/4)*Integral((a
*x**2/b + 1)**(-1/4), (x, 1/x))/(15*a**4*e**6*(a + b*x**2)**(1/4)) - 16*b*sqrt(e
*x)*(9*a*d - 14*b*c)/(15*a**4*e**6*x*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.462106, size = 171, normalized size = 0.78 \[ -\frac{2 \sqrt{e x} \left (a^4 \left (5 c+9 d x^2\right )-2 a^3 b x^2 \left (7 c+45 d x^2\right )+4 a^2 b^2 x^4 \left (35 c-81 d x^2\right )+72 a b^3 x^6 \left (7 c-3 d x^2\right )+16 b^2 x^6 \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (9 a d-14 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+336 b^4 c x^8\right )}{45 a^5 e^6 x^5 \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(9/4)),x]

[Out]

(-2*Sqrt[e*x]*(336*b^4*c*x^8 + 4*a^2*b^2*x^4*(35*c - 81*d*x^2) + 72*a*b^3*x^6*(7
*c - 3*d*x^2) + a^4*(5*c + 9*d*x^2) - 2*a^3*b*x^2*(7*c + 45*d*x^2) + 16*b^2*(-14
*b*c + 9*a*d)*x^6*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4,
7/4, -((b*x^2)/a)]))/(45*a^5*e^6*x^5*(a + b*x^2)^(5/4))

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Maple [F]  time = 0.107, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{11}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(11/2)/(b*x^2+a)^(9/4),x)

[Out]

int((d*x^2+c)/(e*x)^(11/2)/(b*x^2+a)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(11/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(11/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b^{2} e^{5} x^{9} + 2 \, a b e^{5} x^{7} + a^{2} e^{5} x^{5}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(11/2)),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)/((b^2*e^5*x^9 + 2*a*b*e^5*x^7 + a^2*e^5*x^5)*(b*x^2 + a)^(1
/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)**(11/2)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(11/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(11/2)), x)