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

Optimal. Leaf size=211 \[ -\frac{b^{3/4} (4 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}-\frac{b^{3/4} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}+\frac{(b c-a d)^{7/4} \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2}+\frac{(b c-a d)^{7/4} \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2}+\frac{b x \left (a+b x^4\right )^{3/4}}{4 d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.523448, antiderivative size = 211, 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 \[ -\frac{b^{3/4} (4 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}-\frac{b^{3/4} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}+\frac{(b c-a d)^{7/4} \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2}+\frac{(b c-a d)^{7/4} \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2}+\frac{b x \left (a+b x^4\right )^{3/4}}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 62.9504, size = 192, normalized size = 0.91 \[ \frac{b^{\frac{3}{4}} \left (7 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 d^{2}} + \frac{b^{\frac{3}{4}} \left (7 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 d^{2}} + \frac{b x \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 d} + \frac{\left (- a d + b c\right )^{\frac{7}{4}} \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} d^{2}} + \frac{\left (- a d + b c\right )^{\frac{7}{4}} \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 1.80635, size = 396, normalized size = 1.88 \[ \frac{1}{80} \left (\frac{5 \left (4 a^2 d \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+4 b c^{3/4} x \left (a+b x^4\right )^{3/4} \sqrt [4]{b c-a d}+a (b c-4 a d) \log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )-a b c \log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 a (4 a d-b c) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{c^{3/4} d \sqrt [4]{b c-a d}}-\frac{36 a b c x^5 (7 a d-4 b c) F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{d \sqrt [4]{a+b x^4} \left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((-36*a*b*c*(-4*b*c + 7*a*d)*x^5*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x
^4)/c)])/(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*AppellF1[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)]))) +
(5*(4*b*c^(3/4)*(b*c - a*d)^(1/4)*x*(a + b*x^4)^(3/4) + 2*a*(-(b*c) + 4*a*d)*Ar
cTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] + a*(b*c - 4*a*d)*Log[c^
(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] - a*b*c*Log[c^(1/4) + ((b*c - a
*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + 4*a^2*d*Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b
+ a*x^4)^(1/4)]))/(c^(3/4)*d*(b*c - a*d)^(1/4)))/80

_______________________________________________________________________________________

Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 2.72271, size = 2776, normalized size = 13.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(4*(b*x^4 + a)^(3/4)*b*x - 16*d*((b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*
d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d
^6 - a^7*d^7)/(c^3*d^8))^(1/4)*arctan(c^2*d^6*x*((b^7*c^7 - 7*a*b^6*c^6*d + 21*a
^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 +
7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(3/4)/(x*sqrt(((b^7*c^8*d^4 - 7*a*b^6*c^7*d^
5 + 21*a^2*b^5*c^6*d^6 - 35*a^3*b^4*c^5*d^7 + 35*a^4*b^3*c^4*d^8 - 21*a^5*b^2*c^
3*d^9 + 7*a^6*b*c^2*d^10 - a^7*c*d^11)*x^2*sqrt((b^7*c^7 - 7*a*b^6*c^6*d + 21*a^
2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7
*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8)) + (b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^
8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^
6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10*a^9*b*c*d^9 + a^10
*d^10)*sqrt(b*x^4 + a))/x^2) - (b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 1
0*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*(b*x^4 + a)^(1/4))) + 4*d*((256*b^7
*c^4 - 1792*a*b^6*c^3*d + 4704*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b
^3*d^4)/d^8)^(1/4)*arctan(d^6*x*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4704*a^2*b^5*
c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(3/4)/(x*sqrt(((256*b^7*c^
4*d^4 - 1792*a*b^6*c^3*d^5 + 4704*a^2*b^5*c^2*d^6 - 5488*a^3*b^4*c*d^7 + 2401*a^
4*b^3*d^8)*x^2*sqrt((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4704*a^2*b^5*c^2*d^2 - 548
8*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8) + (4096*b^10*c^6 - 43008*a*b^9*c^5*d +
188160*a^2*b^8*c^4*d^2 - 439040*a^3*b^7*c^3*d^3 + 576240*a^4*b^6*c^2*d^4 - 40336
8*a^5*b^5*c*d^5 + 117649*a^6*b^4*d^6)*sqrt(b*x^4 + a))/x^2) - (64*b^5*c^3 - 336*
a*b^4*c^2*d + 588*a^2*b^3*c*d^2 - 343*a^3*b^2*d^3)*(b*x^4 + a)^(1/4))) + 4*d*((b
^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^
3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(1/4)*log(-(c^2
*d^6*x*((b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*
a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(3/4)
+ (b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*
c*d^4 - a^5*d^5)*(b*x^4 + a)^(1/4))/x) - 4*d*((b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*
b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a
^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(1/4)*log((c^2*d^6*x*((b^7*c^7 - 7*a*b^6*c^6*d
+ 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*
d^5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(3/4) - (b^5*c^5 - 5*a*b^4*c^4*d + 10*
a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*(b*x^4 + a)^(1/4
))/x) - d*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4704*a^2*b^5*c^2*d^2 - 5488*a^3*b^4
*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(1/4)*log(-(d^6*x*((256*b^7*c^4 - 1792*a*b^6*c^3
*d + 4704*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(3/4) +
(64*b^5*c^3 - 336*a*b^4*c^2*d + 588*a^2*b^3*c*d^2 - 343*a^3*b^2*d^3)*(b*x^4 + a)
^(1/4))/x) + d*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4704*a^2*b^5*c^2*d^2 - 5488*a^
3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(1/4)*log((d^6*x*((256*b^7*c^4 - 1792*a*b^6
*c^3*d + 4704*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(3/4
) - (64*b^5*c^3 - 336*a*b^4*c^2*d + 588*a^2*b^3*c*d^2 - 343*a^3*b^2*d^3)*(b*x^4
+ a)^(1/4))/x))/d

_______________________________________________________________________________________

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),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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