3.1128 \(\int \frac{c+d x^2}{\sqrt{e x} \left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=79 \[ \frac{2 \sqrt{e x} (a d+4 b c)}{5 a^2 b e \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{e x} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

[Out]

(2*(b*c - a*d)*Sqrt[e*x])/(5*a*b*e*(a + b*x^2)^(5/4)) + (2*(4*b*c + a*d)*Sqrt[e*
x])/(5*a^2*b*e*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.128344, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 \sqrt{e x} (a d+4 b c)}{5 a^2 b e \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{e x} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*c - a*d)*Sqrt[e*x])/(5*a*b*e*(a + b*x^2)^(5/4)) + (2*(4*b*c + a*d)*Sqrt[e*
x])/(5*a^2*b*e*(a + b*x^2)^(1/4))

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Rubi in Sympy [A]  time = 16.3389, size = 90, normalized size = 1.14 \[ - \frac{d \sqrt{e x}}{2 b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{\sqrt{e x} \left (a d + 4 b c\right )}{10 a b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \sqrt{e x} \left (a d + 4 b c\right )}{5 a^{2} b e \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(1/2)/(b*x**2+a)**(9/4),x)

[Out]

-d*sqrt(e*x)/(2*b*e*(a + b*x**2)**(5/4)) + sqrt(e*x)*(a*d + 4*b*c)/(10*a*b*e*(a
+ b*x**2)**(5/4)) + 2*sqrt(e*x)*(a*d + 4*b*c)/(5*a**2*b*e*(a + b*x**2)**(1/4))

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Mathematica [A]  time = 0.0539472, size = 44, normalized size = 0.56 \[ \frac{2 x \left (5 a c+a d x^2+4 b c x^2\right )}{5 a^2 \sqrt{e x} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(5*a*c + 4*b*c*x^2 + a*d*x^2))/(5*a^2*Sqrt[e*x]*(a + b*x^2)^(5/4))

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Maple [A]  time = 0.007, size = 39, normalized size = 0.5 \[{\frac{2\,x \left ( ad{x}^{2}+4\,c{x}^{2}b+5\,ac \right ) }{5\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(9/4),x)

[Out]

2/5*x*(a*d*x^2+4*b*c*x^2+5*a*c)/(b*x^2+a)^(5/4)/a^2/(e*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.238873, size = 84, normalized size = 1.06 \[ \frac{2 \,{\left ({\left (4 \, b c + a d\right )} x^{2} + 5 \, a c\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{5 \,{\left (a^{2} b^{2} e x^{4} + 2 \, a^{3} b e x^{2} + a^{4} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**(1/2)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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