3.1680 \(\int \frac{(c+d x)^{5/4}}{\sqrt [4]{a+b x}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{5 (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{9/4} d^{3/4}}+\frac{5 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{9/4} d^{3/4}}+\frac{5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)}{8 b^2}+\frac{(a+b x)^{3/4} (c+d x)^{5/4}}{2 b} \]

[Out]

(5*(b*c - a*d)*(a + b*x)^(3/4)*(c + d*x)^(1/4))/(8*b^2) + ((a + b*x)^(3/4)*(c +
d*x)^(5/4))/(2*b) - (5*(b*c - a*d)^2*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(
c + d*x)^(1/4))])/(16*b^(9/4)*d^(3/4)) + (5*(b*c - a*d)^2*ArcTanh[(d^(1/4)*(a +
b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(9/4)*d^(3/4))

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Rubi [A]  time = 0.194379, antiderivative size = 167, 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 \[ -\frac{5 (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{9/4} d^{3/4}}+\frac{5 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{9/4} d^{3/4}}+\frac{5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)}{8 b^2}+\frac{(a+b x)^{3/4} (c+d x)^{5/4}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b*c - a*d)*(a + b*x)^(3/4)*(c + d*x)^(1/4))/(8*b^2) + ((a + b*x)^(3/4)*(c +
d*x)^(5/4))/(2*b) - (5*(b*c - a*d)^2*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(
c + d*x)^(1/4))])/(16*b^(9/4)*d^(3/4)) + (5*(b*c - a*d)^2*ArcTanh[(d^(1/4)*(a +
b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(9/4)*d^(3/4))

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Rubi in Sympy [A]  time = 29.0573, size = 151, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{5}{4}}}{2 b} - \frac{5 \left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x} \left (a d - b c\right )}{8 b^{2}} - \frac{5 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{16 b^{\frac{9}{4}} d^{\frac{3}{4}}} + \frac{5 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{16 b^{\frac{9}{4}} d^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(3/4)*(c + d*x)**(5/4)/(2*b) - 5*(a + b*x)**(3/4)*(c + d*x)**(1/4)*(a
*d - b*c)/(8*b**2) - 5*(a*d - b*c)**2*atan(d**(1/4)*(a + b*x)**(1/4)/(b**(1/4)*(
c + d*x)**(1/4)))/(16*b**(9/4)*d**(3/4)) + 5*(a*d - b*c)**2*atanh(d**(1/4)*(a +
b*x)**(1/4)/(b**(1/4)*(c + d*x)**(1/4)))/(16*b**(9/4)*d**(3/4))

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Mathematica [C]  time = 0.196465, size = 111, normalized size = 0.66 \[ \frac{\sqrt [4]{c+d x} \left (5 (b c-a d)^2 \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )-d (a+b x) (5 a d-9 b c-4 b d x)\right )}{8 b^2 d \sqrt [4]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(1/4)*(-(d*(a + b*x)*(-9*b*c + 5*a*d - 4*b*d*x)) + 5*(b*c - a*d)^2*((
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)]))/(8*b^2*d*(a + b*x)^(1/4))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}}{\frac{1}{\sqrt [4]{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.250449, size = 1519, normalized size = 9.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(20*b^2*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^
3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7
+ a^8*d^8)/(b^9*d^3))^(1/4)*arctan((b^3*d*x + a*b^2*d)*((b^8*c^8 - 8*a*b^7*c^7*
d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^
3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b^9*d^3))^(1/4)/((b^2*c^2
- 2*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (b*x + a)*sqrt(((b^4*c
^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*x + a)*
sqrt(d*x + c) + (b^5*d^2*x + a*b^4*d^2)*sqrt((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b
^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a
^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b^9*d^3)))/(b*x + a)))) - 5*b^2*((b^8
*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*
d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b^9*d^
3))^(1/4)*log(5*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4)
+ (b^3*d*x + a*b^2*d)*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b
^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^
7*b*c*d^7 + a^8*d^8)/(b^9*d^3))^(1/4))/(b*x + a)) + 5*b^2*((b^8*c^8 - 8*a*b^7*c^
7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*
c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b^9*d^3))^(1/4)*log(5*(
(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (b^3*d*x + a*b
^2*d)*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a
^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d
^8)/(b^9*d^3))^(1/4))/(b*x + a)) - 4*(4*b*d*x + 9*b*c - 5*a*d)*(b*x + a)^(3/4)*(
d*x + c)^(1/4))/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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