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

Optimal. Leaf size=362 \[ -\frac{6 a^{3/2} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{2 b x (b c-a d)}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (b c-a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}+\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d} \]

[Out]

(6*a*b*x)/(5*d*(a + b*x^2)^(1/4)) - (2*b*(b*c - a*d)*x)/(d^2*(a + b*x^2)^(1/4))
+ (2*b*x*(a + b*x^2)^(3/4))/(5*d) - (6*a^(3/2)*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*Ell
ipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*d*(a + b*x^2)^(1/4)) + (2*Sqrt[a]*S
qrt[b]*(b*c - a*d)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2
, 2])/(d^2*(a + b*x^2)^(1/4)) + (a^(1/4)*(-(b*c) + a*d)^(3/2)*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^(5/2)*x) - (a^(1/4)*(-(b*c) + a*d)^(3/2)*Sqrt[-((b*x^2)/a)]*Elli
pticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)],
-1])/(d^(5/2)*x)

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Rubi [A]  time = 0.683797, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{6 a^{3/2} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{2 b x (b c-a d)}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (b c-a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}+\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

(6*a*b*x)/(5*d*(a + b*x^2)^(1/4)) - (2*b*(b*c - a*d)*x)/(d^2*(a + b*x^2)^(1/4))
+ (2*b*x*(a + b*x^2)^(3/4))/(5*d) - (6*a^(3/2)*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*Ell
ipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*d*(a + b*x^2)^(1/4)) + (2*Sqrt[a]*S
qrt[b]*(b*c - a*d)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2
, 2])/(d^2*(a + b*x^2)^(1/4)) + (a^(1/4)*(-(b*c) + a*d)^(3/2)*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^(5/2)*x) - (a^(1/4)*(-(b*c) + a*d)^(3/2)*Sqrt[-((b*x^2)/a)]*Elli
pticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)],
-1])/(d^(5/2)*x)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - b c\right )^{\frac{3}{2}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d^{\frac{5}{2}} x} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - b c\right )^{\frac{3}{2}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d^{\frac{5}{2}} x} - \frac{3 a^{2} b \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{5 d} + \frac{6 a b x}{5 d \sqrt [4]{a + b x^{2}}} - \frac{a b \left (a d - b c\right ) \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{d^{2}} + \frac{2 b x \left (a + b x^{2}\right )^{\frac{3}{4}}}{5 d} + \frac{2 b x \left (a d - b c\right )}{d^{2} \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*sqrt(-b*x**2/a)*(a*d - b*c)**(3/2)*elliptic_pi(-sqrt(a)*sqrt(d)/sqrt(a*
d - b*c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(d**(5/2)*x) - a**(1/4)*sqrt(-
b*x**2/a)*(a*d - b*c)**(3/2)*elliptic_pi(sqrt(a)*sqrt(d)/sqrt(a*d - b*c), asin((
a + b*x**2)**(1/4)/a**(1/4)), -1)/(d**(5/2)*x) - 3*a**2*b*Integral((a + b*x**2)*
*(-5/4), x)/(5*d) + 6*a*b*x/(5*d*(a + b*x**2)**(1/4)) - a*b*(a*d - b*c)*Integral
((a + b*x**2)**(-5/4), x)/d**2 + 2*b*x*(a + b*x**2)**(3/4)/(5*d) + 2*b*x*(a*d -
b*c)/(d**2*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 1.10477, size = 431, normalized size = 1.19 \[ \frac{2 x \left (\frac{b \left (3 x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-5 a c \left (6 a c+14 a d x^2+b c x^2+6 b d x^4\right ) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}-\frac{9 a^2 c (5 a d-2 b c) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}\right )}{15 d \sqrt [4]{a+b x^2} \left (c+d x^2\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*((-9*a^2*c*(-2*b*c + 5*a*d)*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)])) + (b*(-5*a*c*(6*a*c + b*c*x^2 + 14*a*
d*x^2 + 6*b*d*x^4)*AppellF1[3/2, 1/4, 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, 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)])))/(-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)]))))/(15*d*(a + b*x^2)^(1/4)*(c + d*x^2))

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Maple [F]  time = 0.097, 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((b*x^2+a)^(7/4)/(d*x^2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((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((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. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{7}{4}}}{c + d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x**2)**(7/4)/(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)^(7/4)/(d*x^2 + c),x, algorithm="giac")

[Out]

Exception raised: TypeError