3.1682 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{9/4}} \, dx\)

Optimal. Leaf size=134 \[ -\frac{2 d^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac{2 d^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}} \]

[Out]

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

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Rubi [A]  time = 0.152463, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{2 d^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac{2 d^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/4)/(a + b*x)^(9/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.5671, size = 126, normalized size = 0.94 \[ - \frac{4 \left (c + d x\right )^{\frac{5}{4}}}{5 b \left (a + b x\right )^{\frac{5}{4}}} - \frac{4 d \sqrt [4]{c + d x}}{b^{2} \sqrt [4]{a + b x}} - \frac{2 d^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{b^{\frac{9}{4}}} + \frac{2 d^{\frac{5}{4}} \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{b^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/4)/(b*x+a)**(9/4),x)

[Out]

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

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Mathematica [C]  time = 0.260452, size = 94, normalized size = 0.7 \[ -\frac{4 \sqrt [4]{c+d x} \left (-5 d (a+b x) \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+5 a d+b (c+6 d x)\right )}{5 b^2 (a+b x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/4)/(a + b*x)^(9/4),x]

[Out]

(-4*(c + d*x)^(1/4)*(5*a*d + b*(c + 6*d*x) - 5*d*(a + b*x)*((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)]))/(5
*b^2*(a + b*x)^(5/4))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/4)/(b*x+a)^(9/4),x)

[Out]

int((d*x+c)^(5/4)/(b*x+a)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(9/4), x)

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Fricas [A]  time = 0.242576, size = 468, normalized size = 3.49 \[ -\frac{20 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (b^{3} x + a b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}}}{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} d +{\left (b x + a\right )} \sqrt{\frac{\sqrt{b x + a} \sqrt{d x + c} d^{2} +{\left (b^{5} x + a b^{4}\right )} \sqrt{\frac{d^{5}}{b^{9}}}}{b x + a}}}\right ) - 5 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} d +{\left (b^{3} x + a b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}}}{b x + a}\right ) + 5 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} d -{\left (b^{3} x + a b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}}}{b x + a}\right ) + 4 \,{\left (6 \, b d x + b c + 5 \, a d\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{5 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(9/4),x, algorithm="fricas")

[Out]

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

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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((d*x+c)**(5/4)/(b*x+a)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(9/4),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(9/4), x)