3.937 \(\int (c x)^{5/2} \sqrt [4]{a-b x^2} \, dx\)

Optimal. Leaf size=343 \[ \frac{3 a^2 c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{64 \sqrt{2} b^{7/4}}-\frac{3 a^2 c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{64 \sqrt{2} b^{7/4}}-\frac{3 a^2 c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt{2} b^{7/4}}+\frac{3 a^2 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{32 \sqrt{2} b^{7/4}}+\frac{(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}-\frac{a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b} \]

[Out]

-(a*c*(c*x)^(3/2)*(a - b*x^2)^(1/4))/(16*b) + ((c*x)^(7/2)*(a - b*x^2)^(1/4))/(4
*c) - (3*a^2*c^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)
^(1/4))])/(32*Sqrt[2]*b^(7/4)) + (3*a^2*c^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))])/(32*Sqrt[2]*b^(7/4)) + (3*a^2*c^(5/2)*Log[S
qrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a -
b*x^2)^(1/4)])/(64*Sqrt[2]*b^(7/4)) - (3*a^2*c^(5/2)*Log[Sqrt[c] + (Sqrt[b]*Sqrt
[c]*x)/Sqrt[a - b*x^2] + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a - b*x^2)^(1/4)])/(64*Sqr
t[2]*b^(7/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.845926, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{3 a^2 c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{64 \sqrt{2} b^{7/4}}-\frac{3 a^2 c^{5/2} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{64 \sqrt{2} b^{7/4}}-\frac{3 a^2 c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt{2} b^{7/4}}+\frac{3 a^2 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{32 \sqrt{2} b^{7/4}}+\frac{(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}-\frac{a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(5/2)*(a - b*x^2)^(1/4),x]

[Out]

-(a*c*(c*x)^(3/2)*(a - b*x^2)^(1/4))/(16*b) + ((c*x)^(7/2)*(a - b*x^2)^(1/4))/(4
*c) - (3*a^2*c^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)
^(1/4))])/(32*Sqrt[2]*b^(7/4)) + (3*a^2*c^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))])/(32*Sqrt[2]*b^(7/4)) + (3*a^2*c^(5/2)*Log[S
qrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a -
b*x^2)^(1/4)])/(64*Sqrt[2]*b^(7/4)) - (3*a^2*c^(5/2)*Log[Sqrt[c] + (Sqrt[b]*Sqrt
[c]*x)/Sqrt[a - b*x^2] + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a - b*x^2)^(1/4)])/(64*Sqr
t[2]*b^(7/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 76.9713, size = 311, normalized size = 0.91 \[ \frac{3 \sqrt{2} a^{2} c^{\frac{5}{2}} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} \sqrt{c} \sqrt{c x}}{\sqrt [4]{a - b x^{2}}} + \frac{\sqrt{b} c x}{\sqrt{a - b x^{2}}} + c \right )}}{128 b^{\frac{7}{4}}} - \frac{3 \sqrt{2} a^{2} c^{\frac{5}{2}} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c} \sqrt{c x}}{\sqrt [4]{a - b x^{2}}} + \frac{\sqrt{b} c x}{\sqrt{a - b x^{2}}} + c \right )}}{128 b^{\frac{7}{4}}} + \frac{3 \sqrt{2} a^{2} c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a - b x^{2}}} - 1 \right )}}{64 b^{\frac{7}{4}}} + \frac{3 \sqrt{2} a^{2} c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a - b x^{2}}} + 1 \right )}}{64 b^{\frac{7}{4}}} - \frac{a c \left (c x\right )^{\frac{3}{2}} \sqrt [4]{a - b x^{2}}}{16 b} + \frac{\left (c x\right )^{\frac{7}{2}} \sqrt [4]{a - b x^{2}}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(5/2)*(-b*x**2+a)**(1/4),x)

[Out]

3*sqrt(2)*a**2*c**(5/2)*log(-sqrt(2)*b**(1/4)*sqrt(c)*sqrt(c*x)/(a - b*x**2)**(1
/4) + sqrt(b)*c*x/sqrt(a - b*x**2) + c)/(128*b**(7/4)) - 3*sqrt(2)*a**2*c**(5/2)
*log(sqrt(2)*b**(1/4)*sqrt(c)*sqrt(c*x)/(a - b*x**2)**(1/4) + sqrt(b)*c*x/sqrt(a
- b*x**2) + c)/(128*b**(7/4)) + 3*sqrt(2)*a**2*c**(5/2)*atan(sqrt(2)*b**(1/4)*s
qrt(c*x)/(sqrt(c)*(a - b*x**2)**(1/4)) - 1)/(64*b**(7/4)) + 3*sqrt(2)*a**2*c**(5
/2)*atan(sqrt(2)*b**(1/4)*sqrt(c*x)/(sqrt(c)*(a - b*x**2)**(1/4)) + 1)/(64*b**(7
/4)) - a*c*(c*x)**(3/2)*(a - b*x**2)**(1/4)/(16*b) + (c*x)**(7/2)*(a - b*x**2)**
(1/4)/(4*c)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0671904, size = 84, normalized size = 0.24 \[ -\frac{c (c x)^{3/2} \left (-a^2 \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right )+a^2-5 a b x^2+4 b^2 x^4\right )}{16 b \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(5/2)*(a - b*x^2)^(1/4),x]

[Out]

-(c*(c*x)^(3/2)*(a^2 - 5*a*b*x^2 + 4*b^2*x^4 - a^2*(1 - (b*x^2)/a)^(3/4)*Hyperge
ometric2F1[3/4, 3/4, 7/4, (b*x^2)/a]))/(16*b*(a - b*x^2)^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{5}{2}}}\sqrt [4]{-b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(5/2)*(-b*x^2+a)^(1/4),x)

[Out]

int((c*x)^(5/2)*(-b*x^2+a)^(1/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)*(c*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)*(c*x)^(5/2),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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)**(5/2)*(-b*x**2+a)**(1/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.26256, size = 533, normalized size = 1.55 \[ -\frac{1}{128} \, a^{2} c^{6}{\left (\frac{6 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} + \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b^{\frac{7}{4}} c^{4}} + \frac{6 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} - \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b^{\frac{7}{4}} c^{4}} + \frac{3 \, \sqrt{2} \sqrt{{\left | c \right |}}{\rm ln}\left (\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right )}{b^{\frac{7}{4}} c^{4}} - \frac{3 \, \sqrt{2} \sqrt{{\left | c \right |}}{\rm ln}\left (-\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right )}{b^{\frac{7}{4}} c^{4}} - \frac{8 \,{\left (\frac{3 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b c^{2} \sqrt{{\left | c \right |}}}{\sqrt{c x}} + \frac{{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}}{\left (b c^{2} - \frac{a c^{2}}{x^{2}}\right )} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )} x^{4}}{a^{2} b c^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)*(c*x)^(5/2),x, algorithm="giac")

[Out]

-1/128*a^2*c^6*(6*sqrt(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4)*sqrt(
abs(c)) + 2*(-b*c^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/(b^(1/4)*sqrt(abs
(c))))/(b^(7/4)*c^4) + 6*sqrt(2)*sqrt(abs(c))*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/
4)*sqrt(abs(c)) - 2*(-b*c^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/(b^(1/4)*
sqrt(abs(c))))/(b^(7/4)*c^4) + 3*sqrt(2)*sqrt(abs(c))*ln(sqrt(2)*(-b*c^2*x^2 + a
*c^2)^(1/4)*b^(1/4)*abs(c)/sqrt(c*x) + sqrt(b)*abs(c) + sqrt(-b*c^2*x^2 + a*c^2)
*abs(c)/(c*x))/(b^(7/4)*c^4) - 3*sqrt(2)*sqrt(abs(c))*ln(-sqrt(2)*(-b*c^2*x^2 +
a*c^2)^(1/4)*b^(1/4)*abs(c)/sqrt(c*x) + sqrt(b)*abs(c) + sqrt(-b*c^2*x^2 + a*c^2
)*abs(c)/(c*x))/(b^(7/4)*c^4) - 8*(3*(-b*c^2*x^2 + a*c^2)^(1/4)*b*c^2*sqrt(abs(c
))/sqrt(c*x) + (-b*c^2*x^2 + a*c^2)^(1/4)*(b*c^2 - a*c^2/x^2)*sqrt(abs(c))/sqrt(
c*x))*x^4/(a^2*b*c^6))