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

Optimal. Leaf size=302 \[ \frac{2 a^{3/2} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 d \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} (b c-a d) F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{2 b x \sqrt [4]{a+b x^2}}{3 d} \]

[Out]

(2*b*x*(a + b*x^2)^(1/4))/(3*d) + (2*a^(3/2)*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*Ellip
ticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*d*(a + b*x^2)^(3/4)) - (2*Sqrt[a]*Sqr
t[b]*(b*c - a*d)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2,
2])/(d^2*(a + b*x^2)^(3/4)) + (a^(1/4)*(b*c - a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi
[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1]
)/(d^2*x) + (a^(1/4)*(b*c - a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])
/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^2*x)

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Rubi [A]  time = 0.580659, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{2 a^{3/2} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 d \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} (b c-a d) F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{2 b x \sqrt [4]{a+b x^2}}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

(2*b*x*(a + b*x^2)^(1/4))/(3*d) + (2*a^(3/2)*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*Ellip
ticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*d*(a + b*x^2)^(3/4)) - (2*Sqrt[a]*Sqr
t[b]*(b*c - a*d)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2,
2])/(d^2*(a + b*x^2)^(3/4)) + (a^(1/4)*(b*c - a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi
[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1]
)/(d^2*x) + (a^(1/4)*(b*c - a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])
/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(d^2*x)

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Rubi in Sympy [A]  time = 98.3325, size = 270, normalized size = 0.89 \[ - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - b c\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d^{2} x} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - b c\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d^{2} x} + \frac{2 a^{\frac{3}{2}} \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 d \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \sqrt{a} \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} \left (a d - b c\right ) F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{d^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 b x \sqrt [4]{a + b x^{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/4)*sqrt(-b*x**2/a)*(a*d - b*c)*elliptic_pi(-sqrt(a)*sqrt(d)/sqrt(a*d - b*
c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(d**2*x) - a**(1/4)*sqrt(-b*x**2/a)*
(a*d - b*c)*elliptic_pi(sqrt(a)*sqrt(d)/sqrt(a*d - b*c), asin((a + b*x**2)**(1/4
)/a**(1/4)), -1)/(d**2*x) + 2*a**(3/2)*sqrt(b)*(1 + b*x**2/a)**(3/4)*elliptic_f(
atan(sqrt(b)*x/sqrt(a))/2, 2)/(3*d*(a + b*x**2)**(3/4)) + 2*sqrt(a)*sqrt(b)*(1 +
b*x**2/a)**(3/4)*(a*d - b*c)*elliptic_f(atan(sqrt(b)*x/sqrt(a))/2, 2)/(d**2*(a
+ b*x**2)**(3/4)) + 2*b*x*(a + b*x**2)**(1/4)/(3*d)

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Mathematica [C]  time = 1.11216, size = 435, normalized size = 1.44 \[ \frac{2 x \left (\frac{b \left (3 x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-5 a c \left (6 a c+10 a d x^2+3 b c x^2+6 b d x^4\right ) F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}-\frac{9 a^2 c (3 a d-2 b c) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}\right )}{9 d \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*((-9*a^2*c*(-2*b*c + 3*a*d)*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x
^2)/c)])/(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4
*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2
, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) + (b*(-5*a*c*(6*a*c + 3*b*c*x^2 + 1
0*a*d*x^2 + 6*b*d*x^4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] +
3*x^2*(a + b*x^2)*(c + d*x^2)*(4*a*d*AppellF1[5/2, 3/4, 2, 7/2, -((b*x^2)/a), -(
(d*x^2)/c)] + 3*b*c*AppellF1[5/2, 7/4, 1, 7/2, -((b*x^2)/a), -((d*x^2)/c)])))/(-
10*a*c*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*Appel
lF1[5/2, 3/4, 2, 7/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[5/2, 7/4, 1,
7/2, -((b*x^2)/a), -((d*x^2)/c)]))))/(9*d*(a + b*x^2)^(3/4)*(c + d*x^2))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^(5/4)/(d*x^2 + c),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x**2)**(5/4)/(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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