3.1727 \(\int \frac{1}{\sqrt [4]{a+b x} (c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=719 \[ -\frac{\sqrt{2} \sqrt [4]{b} \sqrt{b c-a d} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{d^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{2 \sqrt{2} \sqrt [4]{b} \sqrt{b c-a d} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{d^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{4 \sqrt{b} \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{\sqrt{d} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^2 (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}+\frac{4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)} \]

[Out]

(4*(a + b*x)^(3/4))/((b*c - a*d)*(c + d*x)^(1/4)) - (4*Sqrt[b]*Sqrt[(a + b*x)*(c
+ d*x)]*Sqrt[(b*c + a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(Sqrt[d]*(
b*c - a*d)^2*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*(1 + (2*Sqrt[
b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))) + (2*Sqrt[2]*b^(1/4)*Sqrt[b*
c - a*d]*((a + b*x)*(c + d*x))^(1/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[
b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/
((b*c - a*d)^2*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2
)]*EllipticE[2*ArcTan[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt
[b*c - a*d]], 1/2])/(d^(3/4)*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*
x)*Sqrt[(a*d + b*(c + 2*d*x))^2]) - (Sqrt[2]*b^(1/4)*Sqrt[b*c - a*d]*((a + b*x)*
(c + d*x))^(1/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(a +
b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/((b*c - a*d)^2*(1 +
(2*Sqrt[b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]*EllipticF[2*ArcTa
n[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt[b*c - a*d]], 1/2])/
(d^(3/4)*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d + b*(c
+ 2*d*x))^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.43092, antiderivative size = 719, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{\sqrt{2} \sqrt [4]{b} \sqrt{b c-a d} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{d^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{2 \sqrt{2} \sqrt [4]{b} \sqrt{b c-a d} \sqrt [4]{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{d^{3/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{4 \sqrt{b} \sqrt{(a+b x) (c+d x)} \sqrt{(a d+b c+2 b d x)^2} \sqrt{(a d+b (c+2 d x))^2}}{\sqrt{d} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^2 (a d+b c+2 b d x) \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )}+\frac{4 (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a + b*x)^(1/4)*(c + d*x)^(5/4)),x]

[Out]

(4*(a + b*x)^(3/4))/((b*c - a*d)*(c + d*x)^(1/4)) - (4*Sqrt[b]*Sqrt[(a + b*x)*(c
+ d*x)]*Sqrt[(b*c + a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(Sqrt[d]*(
b*c - a*d)^2*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*(1 + (2*Sqrt[
b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))) + (2*Sqrt[2]*b^(1/4)*Sqrt[b*
c - a*d]*((a + b*x)*(c + d*x))^(1/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[
b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/
((b*c - a*d)^2*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2
)]*EllipticE[2*ArcTan[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt
[b*c - a*d]], 1/2])/(d^(3/4)*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*
x)*Sqrt[(a*d + b*(c + 2*d*x))^2]) - (Sqrt[2]*b^(1/4)*Sqrt[b*c - a*d]*((a + b*x)*
(c + d*x))^(1/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[b]*Sqrt[d]*Sqrt[(a +
b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/((b*c - a*d)^2*(1 +
(2*Sqrt[b]*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]*EllipticF[2*ArcTa
n[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt[b*c - a*d]], 1/2])/
(d^(3/4)*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d + b*(c
+ 2*d*x))^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 131.125, size = 857, normalized size = 1.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/4)/(d*x+c)**(5/4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.202932, size = 100, normalized size = 0.14 \[ \frac{12 d (a+b x)-8 b (c+d x) \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{3 d \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(1/4)*(c + d*x)^(5/4)),x]

[Out]

(12*d*(a + b*x) - 8*b*((d*(a + b*x))/(-(b*c) + a*d))^(1/4)*(c + d*x)*Hypergeomet
ric2F1[1/4, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/(3*d*(b*c - a*d)*(a + b*x)^(1/
4)*(c + d*x)^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [4]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/4)/(d*x+c)^(5/4),x)

[Out]

int(1/(b*x+a)^(1/4)/(d*x+c)^(5/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/4)*(d*x + c)^(5/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/4)*(d*x + c)^(5/4)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/4)*(d*x + c)^(5/4)),x, algorithm="fricas")

[Out]

integral(1/((b*x + a)^(1/4)*(d*x + c)^(5/4)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/4)/(d*x+c)**(5/4),x)

[Out]

Integral(1/((a + b*x)**(1/4)*(c + d*x)**(5/4)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/4)*(d*x + c)^(5/4)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/4)*(d*x + c)^(5/4)), x)