3.1714 \(\int \frac{\sqrt [4]{a+b x}}{(c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=295 \[ \frac{2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{d}-\frac{(b c-a d)^{3/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{b} d^{5/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}} \]

[Out]

(2*(a + b*x)^(1/4)*(c + d*x)^(1/4))/d - ((b*c - a*d)^(3/2)*((a + b*x)*(c + d*x))
^(3/4)*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*ArcTan[(Sqrt[2]
*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt[b*c - a*d]], 1/2])/(Sqrt[2]*b
^(1/4)*d^(5/4)*(a + b*x)^(3/4)*(c + d*x)^(3/4)*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d +
b*(c + 2*d*x))^2])

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Rubi [A]  time = 0.578724, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{d}-\frac{(b c-a d)^{3/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{b} d^{5/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x)^(1/4)/(c + d*x)^(3/4),x]

[Out]

(2*(a + b*x)^(1/4)*(c + d*x)^(1/4))/d - ((b*c - a*d)^(3/2)*((a + b*x)*(c + d*x))
^(3/4)*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*ArcTan[(Sqrt[2]
*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1/4))/Sqrt[b*c - a*d]], 1/2])/(Sqrt[2]*b
^(1/4)*d^(5/4)*(a + b*x)^(3/4)*(c + d*x)^(3/4)*(b*c + a*d + 2*b*d*x)*Sqrt[(a*d +
b*(c + 2*d*x))^2])

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Rubi in Sympy [A]  time = 45.1414, size = 348, normalized size = 1.18 \[ \frac{2 \sqrt [4]{a + b x} \sqrt [4]{c + d x}}{d} + \frac{\sqrt{2} \sqrt{\frac{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}}{\left (a d - b c\right )^{2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{3}{2}} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}}{a d - b c} + 1\right ) \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{3}{4}} \sqrt{\left (a d + b c + 2 b d x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{a c + b d x^{2} + x \left (a d + b c\right )}}{\sqrt{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt [4]{b} d^{\frac{5}{4}} \left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{3}{4}} \sqrt{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}} \left (a d + b c + 2 b d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/4)/(d*x+c)**(3/4),x)

[Out]

2*(a + b*x)**(1/4)*(c + d*x)**(1/4)/d + sqrt(2)*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))*(a*d - b*c)**(3/2)*(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))**(3/4)*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)
/(2*b**(1/4)*d**(5/4)*(a + b*x)**(3/4)*(c + d*x)**(3/4)*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))

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Mathematica [C]  time = 0.163668, size = 74, normalized size = 0.25 \[ \frac{2 \sqrt [4]{a+b x} \sqrt [4]{c+d x} \left (\frac{\, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [4]{\frac{d (a+b x)}{a d-b c}}}+1\right )}{d} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(1/4)/(c + d*x)^(3/4),x]

[Out]

(2*(a + b*x)^(1/4)*(c + d*x)^(1/4)*(1 + Hypergeometric2F1[1/4, 3/4, 5/4, (b*(c +
d*x))/(b*c - a*d)]/((d*(a + b*x))/(-(b*c) + a*d))^(1/4)))/d

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/4)/(d*x+c)^(3/4),x)

[Out]

int((b*x+a)^(1/4)/(d*x+c)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/(d*x + c)^(3/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)/(d*x + c)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/(d*x + c)^(3/4),x, algorithm="fricas")

[Out]

integral((b*x + a)^(1/4)/(d*x + c)^(3/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x}}{\left (c + d x\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/4)/(d*x+c)**(3/4),x)

[Out]

Integral((a + b*x)**(1/4)/(c + d*x)**(3/4), x)

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GIAC/XCAS [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 + a)^(1/4)/(d*x + c)^(3/4),x, algorithm="giac")

[Out]

Timed out