3.1114 \(\int \frac{c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{5/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.246005, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{4 \sqrt{b} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (6 b c-5 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{5/2} e^4 \sqrt [4]{a+b x^2}}+\frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(5/4)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(7/2)/(b*x**2+a)**(5/4),x)

[Out]

-2*c/(5*a*e*(e*x)**(5/2)*(a + b*x**2)**(1/4)) - 2*(5*a*d - 6*b*c)/(5*a**2*e**3*s
qrt(e*x)*(a + b*x**2)**(1/4)) - 2*sqrt(e*x)*(5*a*d - 6*b*c)*(a/(b*x**2) + 1)**(1
/4)*Integral((a*x**2/b + 1)**(-1/4), (x, 1/x))/(5*a**2*e**4*(a + b*x**2)**(1/4))
+ 4*sqrt(e*x)*(5*a*d - 6*b*c)/(5*a**2*e**4*x*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.178048, size = 114, normalized size = 0.79 \[ \frac{x \left (-6 a^2 \left (c+5 d x^2\right )+8 b x^4 \sqrt [4]{\frac{b x^2}{a}+1} (5 a d-6 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+12 a b \left (3 c x^2-5 d x^4\right )+72 b^2 c x^4\right )}{15 a^3 (e x)^{7/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(5/4)),x]

[Out]

(x*(72*b^2*c*x^4 - 6*a^2*(c + 5*d*x^2) + 12*a*b*(3*c*x^2 - 5*d*x^4) + 8*b*(-6*b*
c + 5*a*d)*x^4*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/
a)]))/(15*a^3*(e*x)^(7/2)*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{7}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(7/2)/(b*x^2+a)^(5/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(7/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b e^{3} x^{5} + a e^{3} x^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(7/2)),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)/((b*e^3*x^5 + a*e^3*x^3)*(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)**(7/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(7/2)),x, algorithm="giac")

[Out]

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