3.111 \(\int \frac{x^2}{a+b (c+d x)^4} \, dx\)

Optimal. Leaf size=318 \[ \frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}+\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{c \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d^3} \]

[Out]

-((c*ArcTan[(Sqrt[b]*(c + d*x)^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*d^3)) - ((Sqrt[a] +
Sqrt[b]*c^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)])/(2*Sqrt[2]*a^(3/4
)*b^(3/4)*d^3) + ((Sqrt[a] + Sqrt[b]*c^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*(c + d*x))
/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)*d^3) + ((Sqrt[a] - Sqrt[b]*c^2)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)
*b^(3/4)*d^3) - ((Sqrt[a] - Sqrt[b]*c^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*(
c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4)*d^3)

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Rubi [A]  time = 0.654407, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588 \[ \frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}-\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}+\frac{\left (\sqrt{a}+\sqrt{b} c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4} d^3}-\frac{c \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*(c + d*x)^4),x]

[Out]

-((c*ArcTan[(Sqrt[b]*(c + d*x)^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*d^3)) - ((Sqrt[a] +
Sqrt[b]*c^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)])/(2*Sqrt[2]*a^(3/4
)*b^(3/4)*d^3) + ((Sqrt[a] + Sqrt[b]*c^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*(c + d*x))
/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)*d^3) + ((Sqrt[a] - Sqrt[b]*c^2)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)
*b^(3/4)*d^3) - ((Sqrt[a] - Sqrt[b]*c^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*(
c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4)*d^3)

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Rubi in Sympy [A]  time = 80.4785, size = 301, normalized size = 0.95 \[ - \frac{c \operatorname{atan}{\left (\frac{\sqrt{b} \left (c + d x\right )^{2}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} d^{3}} + \frac{\sqrt{2} \left (\sqrt{a} - \sqrt{b} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} \left (- c - d x\right ) + \sqrt{a} \sqrt{b} + b \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{3}{4}} d^{3}} - \frac{\sqrt{2} \left (\sqrt{a} - \sqrt{b} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} \left (c + d x\right ) + \sqrt{a} \sqrt{b} + b \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{3}{4}} d^{3}} - \frac{\sqrt{2} \left (\sqrt{a} + \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (- c - d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{3}{4}} d^{3}} + \frac{\sqrt{2} \left (\sqrt{a} + \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{3}{4}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*atan(sqrt(b)*(c + d*x)**2/sqrt(a))/(sqrt(a)*sqrt(b)*d**3) + sqrt(2)*(sqrt(a)
- sqrt(b)*c**2)*log(sqrt(2)*a**(1/4)*b**(3/4)*(-c - d*x) + sqrt(a)*sqrt(b) + b*(
c + d*x)**2)/(8*a**(3/4)*b**(3/4)*d**3) - sqrt(2)*(sqrt(a) - sqrt(b)*c**2)*log(s
qrt(2)*a**(1/4)*b**(3/4)*(c + d*x) + sqrt(a)*sqrt(b) + b*(c + d*x)**2)/(8*a**(3/
4)*b**(3/4)*d**3) - sqrt(2)*(sqrt(a) + sqrt(b)*c**2)*atan(1 + sqrt(2)*b**(1/4)*(
-c - d*x)/a**(1/4))/(4*a**(3/4)*b**(3/4)*d**3) + sqrt(2)*(sqrt(a) + sqrt(b)*c**2
)*atan(1 + sqrt(2)*b**(1/4)*(c + d*x)/a**(1/4))/(4*a**(3/4)*b**(3/4)*d**3)

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Mathematica [C]  time = 0.0534919, size = 106, normalized size = 0.33 \[ \frac{\text{RootSum}\left [\text{$\#$1}^4 b d^4+4 \text{$\#$1}^3 b c d^3+6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1} b c^3 d+a+b c^4\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{\text{$\#$1}^3 d^3+3 \text{$\#$1}^2 c d^2+3 \text{$\#$1} c^2 d+c^3}\&\right ]}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

RootSum[a + b*c^4 + 4*b*c^3*d*#1 + 6*b*c^2*d^2*#1^2 + 4*b*c*d^3*#1^3 + b*d^4*#1^
4 & , (Log[x - #1]*#1^2)/(c^3 + 3*c^2*d*#1 + 3*c*d^2*#1^2 + d^3*#1^3) & ]/(4*b*d
)

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Maple [C]  time = 0.005, size = 97, normalized size = 0.3 \[{\frac{1}{4\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{{d}^{3}{{\it \_R}}^{3}+3\,c{d}^{2}{{\it \_R}}^{2}+3\,{c}^{2}d{\it \_R}+{c}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*(d*x+c)^4),x)

[Out]

1/4/b/d*sum(_R^2/(_R^3*d^3+3*_R^2*c*d^2+3*_R*c^2*d+c^3)*ln(x-_R),_R=RootOf(_Z^4*
b*d^4+4*_Z^3*b*c*d^3+6*_Z^2*b*c^2*d^2+4*_Z*b*c^3*d+b*c^4+a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((d*x + c)^4*b + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 8.28211, size = 274, normalized size = 0.86 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{3} d^{12} + 192 t^{2} a^{2} b^{2} c^{2} d^{6} + t \left (- 32 a^{2} b c d^{3} + 32 a b^{2} c^{5} d^{3}\right ) + a^{2} + 2 a b c^{4} + b^{2} c^{8}, \left ( t \mapsto t \log{\left (x + \frac{64 t^{3} a^{4} b^{2} d^{9} + 448 t^{3} a^{3} b^{3} c^{4} d^{9} + 160 t^{2} a^{3} b^{2} c^{3} d^{6} - 32 t^{2} a^{2} b^{3} c^{7} d^{6} + 60 t a^{3} b c^{2} d^{3} + 256 t a^{2} b^{2} c^{6} d^{3} + 4 t a b^{3} c^{10} d^{3} - 5 a^{3} c - 9 a^{2} b c^{5} - 3 a b^{2} c^{9} + b^{3} c^{13}}{a^{3} d - 33 a^{2} b c^{4} d - 33 a b^{2} c^{8} d + b^{3} c^{12} d} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(256*_t**4*a**3*b**3*d**12 + 192*_t**2*a**2*b**2*c**2*d**6 + _t*(-32*a**2
*b*c*d**3 + 32*a*b**2*c**5*d**3) + a**2 + 2*a*b*c**4 + b**2*c**8, Lambda(_t, _t*
log(x + (64*_t**3*a**4*b**2*d**9 + 448*_t**3*a**3*b**3*c**4*d**9 + 160*_t**2*a**
3*b**2*c**3*d**6 - 32*_t**2*a**2*b**3*c**7*d**6 + 60*_t*a**3*b*c**2*d**3 + 256*_
t*a**2*b**2*c**6*d**3 + 4*_t*a*b**3*c**10*d**3 - 5*a**3*c - 9*a**2*b*c**5 - 3*a*
b**2*c**9 + b**3*c**13)/(a**3*d - 33*a**2*b*c**4*d - 33*a*b**2*c**8*d + b**3*c**
12*d))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((d*x + c)^4*b + a), x)