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

Optimal. Leaf size=309 \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{b x}{2 c d \sqrt [4]{a+b x^2}}+\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac{\sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}} \]

[Out]

-(b*x)/(2*c*d*(a + b*x^2)^(1/4)) + (x*(a + b*x^2)^(3/4))/(2*c*(c + d*x^2)) + (Sq
rt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])
/(2*c*d*(a + b*x^2)^(1/4)) + (a^(1/4)*(b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticP
i[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1
])/(4*c*d^(3/2)*Sqrt[-(b*c) + a*d]*x) - (a^(1/4)*(b*c + 2*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])/(4*c*d^(3/2)*Sqrt[-(b*c) + a*d]*x)

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Rubi [A]  time = 0.618487, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{b x}{2 c d \sqrt [4]{a+b x^2}}+\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac{\sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(b*x)/(2*c*d*(a + b*x^2)^(1/4)) + (x*(a + b*x^2)^(3/4))/(2*c*(c + d*x^2)) + (Sq
rt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])
/(2*c*d*(a + b*x^2)^(1/4)) + (a^(1/4)*(b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticP
i[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1
])/(4*c*d^(3/2)*Sqrt[-(b*c) + a*d]*x) - (a^(1/4)*(b*c + 2*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])/(4*c*d^(3/2)*Sqrt[-(b*c) + a*d]*x)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d + \frac{b c}{2}\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c d^{\frac{3}{2}} x \sqrt{a d - b c}} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d + \frac{b c}{2}\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c d^{\frac{3}{2}} x \sqrt{a d - b c}} + \frac{a b \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{4 c d} - \frac{b x}{2 c d \sqrt [4]{a + b x^{2}}} + \frac{x \left (a + b x^{2}\right )^{\frac{3}{4}}}{2 c \left (c + d x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*sqrt(-b*x**2/a)*(a*d + b*c/2)*elliptic_pi(-sqrt(a)*sqrt(d)/sqrt(a*d - b
*c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(2*c*d**(3/2)*x*sqrt(a*d - b*c)) -
a**(1/4)*sqrt(-b*x**2/a)*(a*d + b*c/2)*elliptic_pi(sqrt(a)*sqrt(d)/sqrt(a*d - b*
c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(2*c*d**(3/2)*x*sqrt(a*d - b*c)) + a
*b*Integral((a + b*x**2)**(-5/4), x)/(4*c*d) - b*x/(2*c*d*(a + b*x**2)**(1/4)) +
x*(a + b*x**2)**(3/4)/(2*c*(c + d*x**2))

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Mathematica [C]  time = 0.293699, size = 320, normalized size = 1.04 \[ \frac{x \left (-\frac{18 a^2 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{5 a b x^2 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{3 \left (a+b x^2\right )}{c}\right )}{6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((3*(a + b*x^2))/c - (18*a^2*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^
2)/c)])/(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*
a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5
/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) + (5*a*b*x^2*AppellF1[3/2, 1/4, 1, 5/2
, -((b*x^2)/a), -((d*x^2)/c)])/(-10*a*c*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a),
-((d*x^2)/c)] + x^2*(4*a*d*AppellF1[5/2, 1/4, 2, 7/2, -((b*x^2)/a), -((d*x^2)/c
)] + b*c*AppellF1[5/2, 5/4, 1, 7/2, -((b*x^2)/a), -((d*x^2)/c)]))))/(6*(a + b*x^
2)^(1/4)*(c + d*x^2))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError