3.919 \(\int (c x)^{7/2} \sqrt [4]{a+b x^2} \, dx\)

Optimal. Leaf size=152 \[ -\frac{a^{5/2} c^2 (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b} \]

[Out]

-(a^2*c^3*Sqrt[c*x]*(a + b*x^2)^(1/4))/(12*b^2) + (a*c*(c*x)^(5/2)*(a + b*x^2)^(
1/4))/(30*b) + ((c*x)^(9/2)*(a + b*x^2)^(1/4))/(5*c) - (a^(5/2)*c^2*(1 + a/(b*x^
2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(12*b^(3/2)*(
a + b*x^2)^(3/4))

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Rubi [A]  time = 0.311851, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{a^{5/2} c^2 (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(7/2)*(a + b*x^2)^(1/4),x]

[Out]

-(a^2*c^3*Sqrt[c*x]*(a + b*x^2)^(1/4))/(12*b^2) + (a*c*(c*x)^(5/2)*(a + b*x^2)^(
1/4))/(30*b) + ((c*x)^(9/2)*(a + b*x^2)^(1/4))/(5*c) - (a^(5/2)*c^2*(1 + a/(b*x^
2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(12*b^(3/2)*(
a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 33.2618, size = 133, normalized size = 0.88 \[ - \frac{a^{\frac{5}{2}} c^{2} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{12 b^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{a^{2} c^{3} \sqrt{c x} \sqrt [4]{a + b x^{2}}}{12 b^{2}} + \frac{a c \left (c x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}}}{30 b} + \frac{\left (c x\right )^{\frac{9}{2}} \sqrt [4]{a + b x^{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(7/2)*(b*x**2+a)**(1/4),x)

[Out]

-a**(5/2)*c**2*(c*x)**(3/2)*(a/(b*x**2) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqr
t(b)*x))/2, 2)/(12*b**(3/2)*(a + b*x**2)**(3/4)) - a**2*c**3*sqrt(c*x)*(a + b*x*
*2)**(1/4)/(12*b**2) + a*c*(c*x)**(5/2)*(a + b*x**2)**(1/4)/(30*b) + (c*x)**(9/2
)*(a + b*x**2)**(1/4)/(5*c)

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Mathematica [C]  time = 0.0680213, size = 98, normalized size = 0.64 \[ \frac{c^3 \sqrt{c x} \left (5 a^3 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-5 a^3-3 a^2 b x^2+14 a b^2 x^4+12 b^3 x^6\right )}{60 b^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(7/2)*(a + b*x^2)^(1/4),x]

[Out]

(c^3*Sqrt[c*x]*(-5*a^3 - 3*a^2*b*x^2 + 14*a*b^2*x^4 + 12*b^3*x^6 + 5*a^3*(1 + (b
*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(60*b^2*(a + b*x
^2)^(3/4))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{7}{2}}}\sqrt [4]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(7/2)*(b*x^2+a)^(1/4),x)

[Out]

int((c*x)^(7/2)*(b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)*(c*x)^(7/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/4)*(c*x)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)*(c*x)^(7/2),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/4)*sqrt(c*x)*c^3*x^3, 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((c*x)**(7/2)*(b*x**2+a)**(1/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)*(c*x)^(7/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/4)*(c*x)^(7/2), x)