3.1718 \(\int \frac{(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{5 \sqrt [4]{b} (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac{5 \sqrt [4]{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}+\frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}} \]

[Out]

(-4*(a + b*x)^(5/4))/(d*(c + d*x)^(1/4)) + (5*b*(a + b*x)^(1/4)*(c + d*x)^(3/4))
/d^2 - (5*b^(1/4)*(b*c - a*d)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x
)^(1/4))])/(2*d^(9/4)) - (5*b^(1/4)*(b*c - a*d)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4)
)/(b^(1/4)*(c + d*x)^(1/4))])/(2*d^(9/4))

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Rubi [A]  time = 0.185936, antiderivative size = 152, 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 \[ -\frac{5 \sqrt [4]{b} (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac{5 \sqrt [4]{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}+\frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(a + b*x)^(5/4))/(d*(c + d*x)^(1/4)) + (5*b*(a + b*x)^(1/4)*(c + d*x)^(3/4))
/d^2 - (5*b^(1/4)*(b*c - a*d)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x
)^(1/4))])/(2*d^(9/4)) - (5*b^(1/4)*(b*c - a*d)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4)
)/(b^(1/4)*(c + d*x)^(1/4))])/(2*d^(9/4))

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Rubi in Sympy [A]  time = 24.5443, size = 141, normalized size = 0.93 \[ \frac{5 \sqrt [4]{b} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{2 d^{\frac{9}{4}}} + \frac{5 \sqrt [4]{b} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{2 d^{\frac{9}{4}}} + \frac{5 b \sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{d^{2}} - \frac{4 \left (a + b x\right )^{\frac{5}{4}}}{d \sqrt [4]{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b**(1/4)*(a*d - b*c)*atan(d**(1/4)*(a + b*x)**(1/4)/(b**(1/4)*(c + d*x)**(1/4)
))/(2*d**(9/4)) + 5*b**(1/4)*(a*d - b*c)*atanh(d**(1/4)*(a + b*x)**(1/4)/(b**(1/
4)*(c + d*x)**(1/4)))/(2*d**(9/4)) + 5*b*(a + b*x)**(1/4)*(c + d*x)**(3/4)/d**2
- 4*(a + b*x)**(5/4)/(d*(c + d*x)**(1/4))

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Mathematica [C]  time = 0.382733, size = 99, normalized size = 0.65 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac{5 b \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [4]{\frac{d (a+b x)}{a d-b c}}}+\frac{3 (-4 a d+5 b c+b d x)}{c+d x}\right )}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/4)*(c + d*x)^(3/4)*((3*(5*b*c - 4*a*d + b*d*x))/(c + d*x) + (5*b*H
ypergeometric2F1[3/4, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)])/((d*(a + b*x))/(-(b*
c) + a*d))^(1/4)))/(3*d^2)

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{4}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243598, size = 930, normalized size = 6.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(20*(d^3*x + c*d^2)*((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b
^2*c*d^3 + a^4*b*d^4)/d^9)^(1/4)*arctan(-(d^3*x + c*d^2)*((b^5*c^4 - 4*a*b^4*c^3
*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)/d^9)^(1/4)/((b*c - a*d)*(b
*x + a)^(1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*
sqrt(b*x + a)*sqrt(d*x + c) + (d^5*x + c*d^4)*sqrt((b^5*c^4 - 4*a*b^4*c^3*d + 6*
a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)/d^9))/(d*x + c)))) + 5*(d^3*x + c
*d^2)*((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^
4)/d^9)^(1/4)*log(-5*((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (d^3*x + c*d
^2)*((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)
/d^9)^(1/4))/(d*x + c)) - 5*(d^3*x + c*d^2)*((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^
3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)/d^9)^(1/4)*log(-5*((b*c - a*d)*(b*x + a
)^(1/4)*(d*x + c)^(3/4) - (d^3*x + c*d^2)*((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*
c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)/d^9)^(1/4))/(d*x + c)) - 4*(b*d*x + 5*b*c
- 4*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(d^3*x + c*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(5/4)/(c + d*x)**(5/4), 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)^(5/4)/(d*x + c)^(5/4),x, algorithm="giac")

[Out]

Timed out