3.1694 \(\int \frac{(a+b x)^{5/4}}{\sqrt [4]{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.20935, 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 \[ \frac{5 (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}}+\frac{5 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}}-\frac{5 \sqrt [4]{a+b x} (c+d x)^{3/4} (b c-a d)}{8 d^2}+\frac{(a+b x)^{5/4} (c+d x)^{3/4}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.726, size = 151, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}}}{2 d} + \frac{5 \sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )}{8 d^{2}} - \frac{5 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{16 b^{\frac{3}{4}} d^{\frac{9}{4}}} + \frac{5 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{16 b^{\frac{3}{4}} d^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.207, size = 108, normalized size = 0.65 \[ \frac{(c+d x)^{3/4} \left (5 (b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+3 d (a+b x) (9 a d-5 b c+4 b d x)\right )}{24 d^3 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(3/4)*(3*d*(a + b*x)*(-5*b*c + 9*a*d + 4*b*d*x) + 5*(b*c - a*d)^2*((d
*(a + b*x))/(-(b*c) + a*d))^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (b*(c + d*x))
/(b*c - a*d)]))/(24*d^3*(a + b*x)^(3/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.248163, size = 1519, normalized size = 9.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(20*d^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^3*d^9))^(1/4)*arctan((b*d^3*x + b*c*d^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^3*d^9))^(1/4)/((b^2*c^2
- 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (d*x + c)*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^2*d^5*x + b^2*c*d^4)*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^3*d^9)))/(d*x + c)))) - 5*d^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^3*d^
9))^(1/4)*log(5*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4)
+ (b*d^3*x + b*c*d^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^3*d^9))^(1/4))/(d*x + c)) + 5*d^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^3*d^9))^(1/4)*log(5*(
(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b*d^3*x + b*c
*d^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^3*d^9))^(1/4))/(d*x + c)) - 4*(4*b*d*x - 5*b*c + 9*a*d)*(b*x + a)^(1/4)*(
d*x + c)^(3/4))/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(5/4)/(c + d*x)**(1/4), 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)^(1/4),x, algorithm="giac")

[Out]

Timed out