3.1115 \(\int \frac{c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=182 \[ \frac{8 b^{3/2} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (10 b c-9 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 a^{7/2} e^6 \sqrt [4]{a+b x^2}}-\frac{4 b (10 b c-9 a d)}{15 a^3 e^5 \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{2 (10 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{9 a e (e x)^{9/2} \sqrt [4]{a+b x^2}} \]

[Out]

(-2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(1/4)) + (2*(10*b*c - 9*a*d))/(45*a^2*e^3*
(e*x)^(5/2)*(a + b*x^2)^(1/4)) - (4*b*(10*b*c - 9*a*d))/(15*a^3*e^5*Sqrt[e*x]*(a
+ b*x^2)^(1/4)) + (8*b^(3/2)*(10*b*c - 9*a*d)*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*E
llipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(15*a^(7/2)*e^6*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.310752, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{8 b^{3/2} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (10 b c-9 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 a^{7/2} e^6 \sqrt [4]{a+b x^2}}-\frac{4 b (10 b c-9 a d)}{15 a^3 e^5 \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{2 (10 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{9 a e (e x)^{9/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(5/4)),x]

[Out]

(-2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(1/4)) + (2*(10*b*c - 9*a*d))/(45*a^2*e^3*
(e*x)^(5/2)*(a + b*x^2)^(1/4)) - (4*b*(10*b*c - 9*a*d))/(15*a^3*e^5*Sqrt[e*x]*(a
+ b*x^2)^(1/4)) + (8*b^(3/2)*(10*b*c - 9*a*d)*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*E
llipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(15*a^(7/2)*e^6*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 c}{9 a e \left (e x\right )^{\frac{9}{2}} \sqrt [4]{a + b x^{2}}} - \frac{2 \left (9 a d - 10 b c\right )}{45 a^{2} e^{3} \left (e x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}}} + \frac{4 b \left (9 a d - 10 b c\right )}{15 a^{3} e^{5} \sqrt{e x} \sqrt [4]{a + b x^{2}}} - \frac{4 b \sqrt{e x} \left (9 a d - 10 b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{15 a^{3} e^{6} \sqrt [4]{a + b x^{2}}} \]

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)**(5/4),x)

[Out]

-2*c/(9*a*e*(e*x)**(9/2)*(a + b*x**2)**(1/4)) - 2*(9*a*d - 10*b*c)/(45*a**2*e**3
*(e*x)**(5/2)*(a + b*x**2)**(1/4)) + 4*b*(9*a*d - 10*b*c)/(15*a**3*e**5*sqrt(e*x
)*(a + b*x**2)**(1/4)) - 4*b*sqrt(e*x)*(9*a*d - 10*b*c)*(a/(b*x**2) + 1)**(1/4)*
Integral((a*x**2/b + 1)**(-5/4), (x, 1/x))/(15*a**3*e**6*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.235699, size = 143, normalized size = 0.79 \[ -\frac{2 \sqrt{e x} \left (a^3 \left (5 c+9 d x^2\right )-2 a^2 b x^2 \left (5 c+27 d x^2\right )+8 b^2 x^6 \sqrt [4]{\frac{b x^2}{a}+1} (9 a d-10 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+12 a b^2 x^4 \left (5 c-9 d x^2\right )+120 b^3 c x^6\right )}{45 a^4 e^6 x^5 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(5/4)),x]

[Out]

(-2*Sqrt[e*x]*(120*b^3*c*x^6 + 12*a*b^2*x^4*(5*c - 9*d*x^2) + a^3*(5*c + 9*d*x^2
) - 2*a^2*b*x^2*(5*c + 27*d*x^2) + 8*b^2*(-10*b*c + 9*a*d)*x^6*(1 + (b*x^2)/a)^(
1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(45*a^4*e^6*x^5*(a + b*x^2
)^(1/4))

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{11}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(11/2)/(b*x^2+a)^(5/4),x)

[Out]

int((d*x^2+c)/(e*x)^(11/2)/(b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(11/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(11/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(11/2)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(11/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(11/2)), x)