3.889 \(\int \frac{x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=201 \[ \frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d} \]

[Out]

-((5*b*c + 7*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(8*b^2*d^2) + (x*(a + b*x)^(1
/4)*(c + d*x)^(3/4))/(2*b*d) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(d^(
1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(11/4)*d^(9/4)) + ((5*b^
2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c +
d*x)^(1/4))])/(16*b^(11/4)*d^(9/4))

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Rubi [A]  time = 0.388967, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

-((5*b*c + 7*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(8*b^2*d^2) + (x*(a + b*x)^(1
/4)*(c + d*x)^(3/4))/(2*b*d) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(d^(
1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(11/4)*d^(9/4)) + ((5*b^
2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c +
d*x)^(1/4))])/(16*b^(11/4)*d^(9/4))

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Rubi in Sympy [A]  time = 30.3991, size = 190, normalized size = 0.95 \[ \frac{x \sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{2 b d} - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (7 a d + 5 b c\right )}{8 b^{2} d^{2}} + \frac{\left (21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{16 b^{\frac{11}{4}} d^{\frac{9}{4}}} + \frac{\left (21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{16 b^{\frac{11}{4}} d^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

x*(a + b*x)**(1/4)*(c + d*x)**(3/4)/(2*b*d) - (a + b*x)**(1/4)*(c + d*x)**(3/4)*
(7*a*d + 5*b*c)/(8*b**2*d**2) + (21*a**2*d**2 + 6*a*b*c*d + 5*b**2*c**2)*atan(d*
*(1/4)*(a + b*x)**(1/4)/(b**(1/4)*(c + d*x)**(1/4)))/(16*b**(11/4)*d**(9/4)) + (
21*a**2*d**2 + 6*a*b*c*d + 5*b**2*c**2)*atanh(d**(1/4)*(a + b*x)**(1/4)/(b**(1/4
)*(c + d*x)**(1/4)))/(16*b**(11/4)*d**(9/4))

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Mathematica [C]  time = 0.257422, size = 123, normalized size = 0.61 \[ \frac{(c+d x)^{3/4} \left (\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-3 d (a+b x) (7 a d+5 b c-4 b d x)\right )}{24 b^2 d^3 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

((c + d*x)^(3/4)*(-3*d*(a + b*x)*(5*b*c + 7*a*d - 4*b*d*x) + (5*b^2*c^2 + 6*a*b*
c*d + 21*a^2*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*b^2*d^3*(a + b*x)^(3/4))

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x + a)^(3/4)*(d*x + c)^(1/4)),x, algorithm="maxima")

[Out]

integrate(x^2/((b*x + a)^(3/4)*(d*x + c)^(1/4)), x)

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Fricas [A]  time = 0.282624, size = 1577, normalized size = 7.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x + a)^(3/4)*(d*x + c)^(1/4)),x, algorithm="fricas")

[Out]

-1/32*(4*b^2*d^2*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 4212
0*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6
*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/4)*arctan((b^
3*d^3*x + b^3*c*d^2)*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 +
42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476
*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/4)/((5*b^
2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (d*x + c)*sqrt
(((25*b^4*c^4 + 60*a*b^3*c^3*d + 246*a^2*b^2*c^2*d^2 + 252*a^3*b*c*d^3 + 441*a^4
*d^4)*sqrt(b*x + a)*sqrt(d*x + c) + (b^6*d^5*x + b^6*c*d^4)*sqrt((625*b^8*c^8 +
3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^
4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7
+ 194481*a^8*d^8)/(b^11*d^9)))/(d*x + c)))) - b^2*d^2*((625*b^8*c^8 + 3000*a*b^
7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4
+ 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481
*a^8*d^8)/(b^11*d^9))^(1/4)*log(((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*(b*x + a)^
(1/4)*(d*x + c)^(3/4) + (b^3*d^3*x + b^3*c*d^2)*((625*b^8*c^8 + 3000*a*b^7*c^7*d
+ 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 1769
04*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^
8)/(b^11*d^9))^(1/4))/(d*x + c)) + b^2*d^2*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15
900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^
5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b
^11*d^9))^(1/4)*log(((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*(b*x + a)^(1/4)*(d*x +
c)^(3/4) - (b^3*d^3*x + b^3*c*d^2)*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2
*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c
^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9
))^(1/4))/(d*x + c)) - 4*(4*b*d*x - 5*b*c - 7*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/
4))/(b^2*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

Integral(x**2/((a + b*x)**(3/4)*(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(x^2/((b*x + a)^(3/4)*(d*x + c)^(1/4)),x, algorithm="giac")

[Out]

Timed out