3.1681 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.179246, 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 \[ -\frac{5 \sqrt [4]{d} (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac{5 \sqrt [4]{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac{5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac{4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.4875, size = 141, normalized size = 0.93 \[ - \frac{4 \left (c + d x\right )^{\frac{5}{4}}}{b \sqrt [4]{a + b x}} + \frac{5 d \left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}{b^{2}} - \frac{5 \sqrt [4]{d} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{2 b^{\frac{9}{4}}} - \frac{5 \sqrt [4]{d} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{2 b^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.20474, size = 93, normalized size = 0.61 \[ \frac{\sqrt [4]{c+d x} \left (5 (b c-a d) \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+5 a d-4 b c+b d x\right )}{b^2 \sqrt [4]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243948, size = 930, normalized size = 6.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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