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

Optimal. Leaf size=230 \[ \frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}+\frac{b^{7/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

[Out]

-((b*c - a*d)*x*(a + b*x^4)^(3/4))/(4*c*d*(c + d*x^4)) + (b^(7/4)*ArcTan[(b^(1/4
)*x)/(a + b*x^4)^(1/4)])/(2*d^2) - ((b*c - a*d)^(3/4)*(4*b*c + 3*a*d)*ArcTan[((b
*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^2) + (b^(7/4)*ArcT
anh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*d^2) - ((b*c - a*d)^(3/4)*(4*b*c + 3*a*d)
*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^2)

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Rubi [A]  time = 0.438932, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}+\frac{b^{7/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x*(a + b*x^4)^(3/4))/(4*c*d*(c + d*x^4)) + (b^(7/4)*ArcTan[(b^(1/4
)*x)/(a + b*x^4)^(1/4)])/(2*d^2) - ((b*c - a*d)^(3/4)*(4*b*c + 3*a*d)*ArcTan[((b
*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^2) + (b^(7/4)*ArcT
anh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*d^2) - ((b*c - a*d)^(3/4)*(4*b*c + 3*a*d)
*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*d^2)

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Rubi in Sympy [A]  time = 63.5151, size = 206, normalized size = 0.9 \[ \frac{b^{\frac{7}{4}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 d^{2}} + \frac{b^{\frac{7}{4}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 d^{2}} + \frac{x \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )}{4 c d \left (c + d x^{4}\right )} - \frac{\left (- a d + b c\right )^{\frac{3}{4}} \left (3 a d + 4 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} d^{2}} - \frac{\left (- a d + b c\right )^{\frac{3}{4}} \left (3 a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(7/4)*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(2*d**2) + b**(7/4)*atanh(b**(1/4)
*x/(a + b*x**4)**(1/4))/(2*d**2) + x*(a + b*x**4)**(3/4)*(a*d - b*c)/(4*c*d*(c +
d*x**4)) - (-a*d + b*c)**(3/4)*(3*a*d + 4*b*c)*atan(x*(-a*d + b*c)**(1/4)/(c**(
1/4)*(a + b*x**4)**(1/4)))/(8*c**(7/4)*d**2) - (-a*d + b*c)**(3/4)*(3*a*d + 4*b*
c)*atanh(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(8*c**(7/4)*d**2)

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Mathematica [C]  time = 0.81934, size = 462, normalized size = 2.01 \[ \frac{3 a^2 \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{16 c^{7/4} \sqrt [4]{b c-a d}}-\frac{9 a b^2 c x^5 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{5 d \sqrt [4]{a+b x^4} \left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+\frac{a b \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{16 c^{3/4} d \sqrt [4]{b c-a d}}-\frac{x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((b*c - a*d)*x*(a + b*x^4)^(3/4))/(4*c*d*(c + d*x^4)) - (9*a*b^2*c*x^5*AppellF1
[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/(5*d*(a + b*x^4)^(1/4)*(c + d*x^
4)*(-9*a*c*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*A
ppellF1[9/4, 1/4, 2, 13/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[9/4, 5/4,
1, 13/4, -((b*x^4)/a), -((d*x^4)/c)]))) + (3*a^2*(2*ArcTan[((b*c - a*d)^(1/4)*x)
/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^
(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(16*c^(7/4)*(b
*c - a*d)^(1/4)) + (a*b*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/
4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((
b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(16*c^(3/4)*d*(b*c - a*d)^(1/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.42465, size = 1972, normalized size = 8.57 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(4*(b*x^4 + a)^(3/4)*(b*c - a*d)*x + 4*(c*d^2*x^4 + c^2*d)*((256*b^7*c^7 -
672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c
^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)*arctan(c^5*d^6*x*((256*b
^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a
^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(3/4)/(x*sqrt(((256*b^
7*c^10*d^4 - 672*a^2*b^5*c^8*d^6 - 112*a^3*b^4*c^7*d^7 + 609*a^4*b^3*c^6*d^8 + 1
89*a^5*b^2*c^5*d^9 - 189*a^6*b*c^4*d^10 - 81*a^7*c^3*d^11)*x^2*sqrt((256*b^7*c^7
- 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2
*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8)) + (4096*b^10*c^10 + 2048*a*b
^9*c^9*d - 14592*a^2*b^8*c^8*d^2 - 9472*a^3*b^7*c^7*d^3 + 18928*a^4*b^6*c^6*d^4
+ 15624*a^5*b^5*c^5*d^5 - 9639*a^6*b^4*c^4*d^6 - 11124*a^7*b^3*c^3*d^7 + 486*a^8
*b^2*c^2*d^8 + 2916*a^9*b*c*d^9 + 729*a^10*d^10)*sqrt(b*x^4 + a))/x^2) + (64*b^5
*c^5 + 16*a*b^4*c^4*d - 116*a^2*b^3*c^3*d^2 - 45*a^3*b^2*c^2*d^3 + 54*a^4*b*c*d^
4 + 27*a^5*d^5)*(b*x^4 + a)^(1/4))) - 16*(c*d^2*x^4 + c^2*d)*(b^7/d^8)^(1/4)*arc
tan(d^6*x*(b^7/d^8)^(3/4)/((b*x^4 + a)^(1/4)*b^5 + x*sqrt((b^7*d^4*x^2*sqrt(b^7/
d^8) + sqrt(b*x^4 + a)*b^10)/x^2))) + (c*d^2*x^4 + c^2*d)*((256*b^7*c^7 - 672*a^
2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5
- 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)*log((c^5*d^6*x*((256*b^7*c^7 -
672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^
2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(3/4) + (64*b^5*c^5 + 16*a*b^4*
c^4*d - 116*a^2*b^3*c^3*d^2 - 45*a^3*b^2*c^2*d^3 + 54*a^4*b*c*d^4 + 27*a^5*d^5)*
(b*x^4 + a)^(1/4))/x) - (c*d^2*x^4 + c^2*d)*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2
- 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*
d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)*log(-(c^5*d^6*x*((256*b^7*c^7 - 672*a^2*b^5*c
^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a
^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(3/4) - (64*b^5*c^5 + 16*a*b^4*c^4*d - 116*a
^2*b^3*c^3*d^2 - 45*a^3*b^2*c^2*d^3 + 54*a^4*b*c*d^4 + 27*a^5*d^5)*(b*x^4 + a)^(
1/4))/x) - 4*(c*d^2*x^4 + c^2*d)*(b^7/d^8)^(1/4)*log((d^6*x*(b^7/d^8)^(3/4) + (b
*x^4 + a)^(1/4)*b^5)/x) + 4*(c*d^2*x^4 + c^2*d)*(b^7/d^8)^(1/4)*log(-(d^6*x*(b^7
/d^8)^(3/4) - (b*x^4 + a)^(1/4)*b^5)/x))/(c*d^2*x^4 + c^2*d)

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Sympy [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**4+a)**(7/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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