3.320 \(\int \frac{\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx\)

Optimal. Leaf size=244 \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{3/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{3/2} x}+\frac{2 b x}{d \sqrt [4]{a+b x^2}}-\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d \sqrt [4]{a+b x^2}} \]

[Out]

(2*b*x)/(d*(a + b*x^2)^(1/4)) - (2*Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*Ellipti
cE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d*(a + b*x^2)^(1/4)) + (a^(1/4)*Sqrt[-(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^(3/2)*x) - (a^(1/4)*Sqrt[-(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^(3/2)*x)

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Rubi [A]  time = 0.455438, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{3/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \sqrt{a d-b c} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{3/2} x}+\frac{2 b x}{d \sqrt [4]{a+b x^2}}-\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/4)/(c + d*x^2),x]

[Out]

(2*b*x)/(d*(a + b*x^2)^(1/4)) - (2*Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*Ellipti
cE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d*(a + b*x^2)^(1/4)) + (a^(1/4)*Sqrt[-(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^(3/2)*x) - (a^(1/4)*Sqrt[-(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^(3/2)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/4)/(d*x**2+c),x)

[Out]

a**(1/4)*sqrt(-b*x**2/a)*sqrt(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**(3/2)*x) - a**(1/4)*sqrt(-b*x
**2/a)*sqrt(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**(3/2)*x) + b*Integral((a + b*x**2)**(-1/4), x)/d

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Mathematica [C]  time = 0.249817, size = 161, normalized size = 0.66 \[ \frac{6 a c x \left (a+b x^2\right )^{3/4} F_1\left (\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (x^2 \left (3 b c F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-4 a d F_1\left (\frac{3}{2};-\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+6 a c F_1\left (\frac{1}{2};-\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^2)^(3/4)/(c + d*x^2),x]

[Out]

(6*a*c*x*(a + b*x^2)^(3/4)*AppellF1[1/2, -3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c
)])/((c + d*x^2)*(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*Ap
pellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/4)/(d*x^2+c),x)

[Out]

int((b*x^2+a)^(3/4)/(d*x^2+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/4)/(d*x^2 + c),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/4)/(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)^(3/4)/(d*x^2 + c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{3}{4}}}{c + d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/4)/(d*x**2+c),x)

[Out]

Integral((a + b*x**2)**(3/4)/(c + d*x**2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/4)/(d*x^2 + c),x, algorithm="giac")

[Out]

Exception raised: TypeError