3.325 \(\int \frac{1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=254 \[ \frac{2 b x}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{2 \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.475576, antiderivative size = 254, 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 \[ \frac{2 b x}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{2 \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 88.9241, size = 223, normalized size = 0.88 \[ - \frac{\sqrt [4]{a} d \sqrt{- \frac{b x^{2}}{a}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{x \left (a d - b c\right )^{2}} - \frac{\sqrt [4]{a} d \sqrt{- \frac{b x^{2}}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{x \left (a d - b c\right )^{2}} - \frac{2 b x}{3 a \left (a + b x^{2}\right )^{\frac{3}{4}} \left (a d - b c\right )} - \frac{2 \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} \left (a + b x^{2}\right )^{\frac{3}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.511052, size = 342, normalized size = 1.35 \[ \frac{2 x \left (\frac{5 b c d x^2 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}+\frac{9 c (b c-3 a d) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}-\frac{3 b}{a}\right )}{9 \left (a+b x^2\right )^{3/4} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*((-3*b)/a + (9*c*(b*c - 3*a*d)*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*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))) + (5*b*c*d*x^2*App
ellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/((c + d*x^2)*(-10*a*c*Appel
lF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 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)])))))/(9*(-(b*c) + a*d)*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x**2)**(7/4)*(c + d*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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