3.872 \(\int \frac{x^2 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=268 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (7 a^2 d^2+10 a b c d+15 b^2 c^2\right )}{32 b^2 d^3}-\frac{(b c-a d) \left (7 a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{11/4} d^{13/4}}-\frac{(b c-a d) \left (7 a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{11/4} d^{13/4}}-\frac{(a+b x)^{5/4} (c+d x)^{3/4} (7 a d+9 b c)}{24 b^2 d^2}+\frac{x (a+b x)^{5/4} (c+d x)^{3/4}}{3 b d} \]

[Out]

((15*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(32*b^2*
d^3) - ((9*b*c + 7*a*d)*(a + b*x)^(5/4)*(c + d*x)^(3/4))/(24*b^2*d^2) + (x*(a +
b*x)^(5/4)*(c + d*x)^(3/4))/(3*b*d) - ((b*c - a*d)*(15*b^2*c^2 + 10*a*b*c*d + 7*
a^2*d^2)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(11/
4)*d^(13/4)) - ((b*c - a*d)*(15*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*ArcTanh[(d^(1/
4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(11/4)*d^(13/4))

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Rubi [A]  time = 0.533199, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (7 a^2 d^2+10 a b c d+15 b^2 c^2\right )}{32 b^2 d^3}-\frac{(b c-a d) \left (7 a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{11/4} d^{13/4}}-\frac{(b c-a d) \left (7 a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{11/4} d^{13/4}}-\frac{(a+b x)^{5/4} (c+d x)^{3/4} (7 a d+9 b c)}{24 b^2 d^2}+\frac{x (a+b x)^{5/4} (c+d x)^{3/4}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(a + b*x)^(1/4))/(c + d*x)^(1/4),x]

[Out]

((15*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(32*b^2*
d^3) - ((9*b*c + 7*a*d)*(a + b*x)^(5/4)*(c + d*x)^(3/4))/(24*b^2*d^2) + (x*(a +
b*x)^(5/4)*(c + d*x)^(3/4))/(3*b*d) - ((b*c - a*d)*(15*b^2*c^2 + 10*a*b*c*d + 7*
a^2*d^2)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(11/
4)*d^(13/4)) - ((b*c - a*d)*(15*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*ArcTanh[(d^(1/
4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(11/4)*d^(13/4))

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Rubi in Sympy [A]  time = 40.2613, size = 255, normalized size = 0.95 \[ \frac{x \left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}}}{3 b d} - \frac{\left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}} \left (7 a d + 9 b c\right )}{24 b^{2} d^{2}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (7 a^{2} d^{2} + 10 a b c d + 15 b^{2} c^{2}\right )}{32 b^{2} d^{3}} + \frac{\left (a d - b c\right ) \left (7 a^{2} d^{2} + 10 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{64 b^{\frac{11}{4}} d^{\frac{13}{4}}} + \frac{\left (a d - b c\right ) \left (7 a^{2} d^{2} + 10 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{64 b^{\frac{11}{4}} d^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**(1/4)/(d*x+c)**(1/4),x)

[Out]

x*(a + b*x)**(5/4)*(c + d*x)**(3/4)/(3*b*d) - (a + b*x)**(5/4)*(c + d*x)**(3/4)*
(7*a*d + 9*b*c)/(24*b**2*d**2) + (a + b*x)**(1/4)*(c + d*x)**(3/4)*(7*a**2*d**2
+ 10*a*b*c*d + 15*b**2*c**2)/(32*b**2*d**3) + (a*d - b*c)*(7*a**2*d**2 + 10*a*b*
c*d + 15*b**2*c**2)*atan(d**(1/4)*(a + b*x)**(1/4)/(b**(1/4)*(c + d*x)**(1/4)))/
(64*b**(11/4)*d**(13/4)) + (a*d - b*c)*(7*a**2*d**2 + 10*a*b*c*d + 15*b**2*c**2)
*atanh(d**(1/4)*(a + b*x)**(1/4)/(b**(1/4)*(c + d*x)**(1/4)))/(64*b**(11/4)*d**(
13/4))

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Mathematica [C]  time = 0.283548, size = 168, normalized size = 0.63 \[ \frac{(c+d x)^{3/4} \left (\left (7 a^3 d^3+3 a^2 b c d^2+5 a b^2 c^2 d-15 b^3 c^3\right ) \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-d (a+b x) \left (7 a^2 d^2+2 a b d (3 c-2 d x)+b^2 \left (-45 c^2+36 c d x-32 d^2 x^2\right )\right )\right )}{96 b^2 d^4 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(a + b*x)^(1/4))/(c + d*x)^(1/4),x]

[Out]

((c + d*x)^(3/4)*(-(d*(a + b*x)*(7*a^2*d^2 + 2*a*b*d*(3*c - 2*d*x) + b^2*(-45*c^
2 + 36*c*d*x - 32*d^2*x^2))) + (-15*b^3*c^3 + 5*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 7*
a^3*d^3)*((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)]))/(96*b^2*d^4*(a + b*x)^(3/4))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{{x}^{2}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(1/4)/(d*x+c)^(1/4),x)

[Out]

int(x^2*(b*x+a)^(1/4)/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}} x^{2}}{{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)*x^2/(d*x + c)^(1/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)*x^2/(d*x + c)^(1/4), x)

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Fricas [A]  time = 0.357895, size = 2259, normalized size = 8.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)*x^2/(d*x + c)^(1/4),x, algorithm="fricas")

[Out]

-1/384*(12*b^2*d^3*((50625*b^12*c^12 - 67500*a*b^11*c^11*d - 6750*a^2*b^10*c^10*
d^2 - 61500*a^3*b^9*c^9*d^3 + 93775*a^4*b^8*c^8*d^4 + 18600*a^5*b^7*c^7*d^5 + 31
580*a^6*b^6*c^6*d^6 - 48600*a^7*b^5*c^5*d^7 - 15249*a^8*b^4*c^4*d^8 - 11004*a^9*
b^3*c^3*d^9 + 9506*a^10*b^2*c^2*d^10 + 4116*a^11*b*c*d^11 + 2401*a^12*d^12)/(b^1
1*d^13))^(1/4)*arctan(-(b^3*d^4*x + b^3*c*d^3)*((50625*b^12*c^12 - 67500*a*b^11*
c^11*d - 6750*a^2*b^10*c^10*d^2 - 61500*a^3*b^9*c^9*d^3 + 93775*a^4*b^8*c^8*d^4
+ 18600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 - 48600*a^7*b^5*c^5*d^7 - 15249*
a^8*b^4*c^4*d^8 - 11004*a^9*b^3*c^3*d^9 + 9506*a^10*b^2*c^2*d^10 + 4116*a^11*b*c
*d^11 + 2401*a^12*d^12)/(b^11*d^13))^(1/4)/((15*b^3*c^3 - 5*a*b^2*c^2*d - 3*a^2*
b*c*d^2 - 7*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((225*b^6*
c^6 - 150*a*b^5*c^5*d - 65*a^2*b^4*c^4*d^2 - 180*a^3*b^3*c^3*d^3 + 79*a^4*b^2*c^
2*d^4 + 42*a^5*b*c*d^5 + 49*a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (b^6*d^7*x +
b^6*c*d^6)*sqrt((50625*b^12*c^12 - 67500*a*b^11*c^11*d - 6750*a^2*b^10*c^10*d^2
- 61500*a^3*b^9*c^9*d^3 + 93775*a^4*b^8*c^8*d^4 + 18600*a^5*b^7*c^7*d^5 + 31580*
a^6*b^6*c^6*d^6 - 48600*a^7*b^5*c^5*d^7 - 15249*a^8*b^4*c^4*d^8 - 11004*a^9*b^3*
c^3*d^9 + 9506*a^10*b^2*c^2*d^10 + 4116*a^11*b*c*d^11 + 2401*a^12*d^12)/(b^11*d^
13)))/(d*x + c)))) + 3*b^2*d^3*((50625*b^12*c^12 - 67500*a*b^11*c^11*d - 6750*a^
2*b^10*c^10*d^2 - 61500*a^3*b^9*c^9*d^3 + 93775*a^4*b^8*c^8*d^4 + 18600*a^5*b^7*
c^7*d^5 + 31580*a^6*b^6*c^6*d^6 - 48600*a^7*b^5*c^5*d^7 - 15249*a^8*b^4*c^4*d^8
- 11004*a^9*b^3*c^3*d^9 + 9506*a^10*b^2*c^2*d^10 + 4116*a^11*b*c*d^11 + 2401*a^1
2*d^12)/(b^11*d^13))^(1/4)*log(-((15*b^3*c^3 - 5*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 7
*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^3*d^4*x + b^3*c*d^3)*((50625*b^12
*c^12 - 67500*a*b^11*c^11*d - 6750*a^2*b^10*c^10*d^2 - 61500*a^3*b^9*c^9*d^3 + 9
3775*a^4*b^8*c^8*d^4 + 18600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 - 48600*a^7
*b^5*c^5*d^7 - 15249*a^8*b^4*c^4*d^8 - 11004*a^9*b^3*c^3*d^9 + 9506*a^10*b^2*c^2
*d^10 + 4116*a^11*b*c*d^11 + 2401*a^12*d^12)/(b^11*d^13))^(1/4))/(d*x + c)) - 3*
b^2*d^3*((50625*b^12*c^12 - 67500*a*b^11*c^11*d - 6750*a^2*b^10*c^10*d^2 - 61500
*a^3*b^9*c^9*d^3 + 93775*a^4*b^8*c^8*d^4 + 18600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6
*c^6*d^6 - 48600*a^7*b^5*c^5*d^7 - 15249*a^8*b^4*c^4*d^8 - 11004*a^9*b^3*c^3*d^9
+ 9506*a^10*b^2*c^2*d^10 + 4116*a^11*b*c*d^11 + 2401*a^12*d^12)/(b^11*d^13))^(1
/4)*log(-((15*b^3*c^3 - 5*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 7*a^3*d^3)*(b*x + a)^(1/
4)*(d*x + c)^(3/4) - (b^3*d^4*x + b^3*c*d^3)*((50625*b^12*c^12 - 67500*a*b^11*c^
11*d - 6750*a^2*b^10*c^10*d^2 - 61500*a^3*b^9*c^9*d^3 + 93775*a^4*b^8*c^8*d^4 +
18600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 - 48600*a^7*b^5*c^5*d^7 - 15249*a^
8*b^4*c^4*d^8 - 11004*a^9*b^3*c^3*d^9 + 9506*a^10*b^2*c^2*d^10 + 4116*a^11*b*c*d
^11 + 2401*a^12*d^12)/(b^11*d^13))^(1/4))/(d*x + c)) - 4*(32*b^2*d^2*x^2 + 45*b^
2*c^2 - 6*a*b*c*d - 7*a^2*d^2 - 4*(9*b^2*c*d - a*b*d^2)*x)*(b*x + a)^(1/4)*(d*x
+ c)^(3/4))/(b^2*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt [4]{a + b x}}{\sqrt [4]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)**(1/4)/(d*x+c)**(1/4),x)

[Out]

Integral(x**2*(a + b*x)**(1/4)/(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)^(1/4)*x^2/(d*x + c)^(1/4),x, algorithm="giac")

[Out]

Timed out