3.115 \(\int \frac{1}{x^2 \left (a+b (c+d x)^4\right )} \, dx\)

Optimal. Leaf size=496 \[ -\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )+\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )+\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{\sqrt{b} c d \left (a-b c^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a+b c^4\right )^2}-\frac{1}{x \left (a+b c^4\right )}-\frac{4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac{b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.92344, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.706 \[ -\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )+\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )+\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{\sqrt{b} c d \left (a-b c^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a+b c^4\right )^2}-\frac{1}{x \left (a+b c^4\right )}-\frac{4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac{b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.19518, size = 238, normalized size = 0.48 \[ \frac{d x \text{RootSum}\left [\text{$\#$1}^4 b d^4+4 \text{$\#$1}^3 b c d^3+6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1} b c^3 d+a+b c^4\&,\frac{4 \text{$\#$1}^3 b c^3 d^3 \log (x-\text{$\#$1})-\text{$\#$1}^2 a d^2 \log (x-\text{$\#$1})+15 \text{$\#$1}^2 b c^4 d^2 \log (x-\text{$\#$1})-6 a c^2 \log (x-\text{$\#$1})-4 \text{$\#$1} a c d \log (x-\text{$\#$1})+10 b c^6 \log (x-\text{$\#$1})+20 \text{$\#$1} b c^5 d \log (x-\text{$\#$1})}{\text{$\#$1}^3 d^3+3 \text{$\#$1}^2 c d^2+3 \text{$\#$1} c^2 d+c^3}\&\right ]-4 \left (a+b c^4+4 b c^3 d x \log (x)\right )}{4 x \left (a+b c^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(a + b*c^4 + 4*b*c^3*d*x*Log[x]) + d*x*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 & , (-6*a*c^2*Log[x - #1] + 10*b*c^
6*Log[x - #1] - 4*a*c*d*Log[x - #1]*#1 + 20*b*c^5*d*Log[x - #1]*#1 - a*d^2*Log[x
- #1]*#1^2 + 15*b*c^4*d^2*Log[x - #1]*#1^2 + 4*b*c^3*d^3*Log[x - #1]*#1^3)/(c^3
+ 3*c^2*d*#1 + 3*c*d^2*#1^2 + d^3*#1^3) & ])/(4*(a + b*c^4)^2*x)

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Maple [C]  time = 0.017, size = 188, normalized size = 0.4 \[ -{\frac{1}{ \left ( b{c}^{4}+a \right ) x}}-4\,{\frac{b{c}^{3}d\ln \left ( x \right ) }{ \left ( b{c}^{4}+a \right ) ^{2}}}+{\frac{d}{4\, \left ( b{c}^{4}+a \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{ \left ( 4\,b{d}^{3}{c}^{3}{{\it \_R}}^{3}+{d}^{2} \left ( 15\,b{c}^{4}-a \right ){{\it \_R}}^{2}+4\,cd \left ( 5\,b{c}^{4}-a \right ){\it \_R}+10\,b{c}^{6}-6\,a{c}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}{d}^{3}+3\,{{\it \_R}}^{2}c{d}^{2}+3\,{\it \_R}\,{c}^{2}d+{c}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*(d*x+c)^4),x)

[Out]

-1/(b*c^4+a)/x-4*b*c^3*d*ln(x)/(b*c^4+a)^2+1/4*d/(b*c^4+a)^2*sum((4*b*d^3*c^3*_R
^3+d^2*(15*b*c^4-a)*_R^2+4*c*d*(5*b*c^4-a)*_R+10*b*c^6-6*a*c^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. \[ -\frac{4 \, b c^{3} d \log \left (x\right )}{b^{2} c^{8} + 2 \, a b c^{4} + a^{2}} + \frac{b d^{2} \int \frac{4 \, b c^{3} d^{3} x^{3} + 10 \, b c^{6} +{\left (15 \, b c^{4} - a\right )} d^{2} x^{2} - 6 \, a c^{2} + 4 \,{\left (5 \, b c^{5} - a c\right )} d x}{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}\,{d x}}{b^{2} c^{8} + 2 \, a b c^{4} + a^{2}} - \frac{1}{{\left (b c^{4} + a\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*b*c^3*d*log(x)/(b^2*c^8 + 2*a*b*c^4 + a^2) + b*d^2*integrate((4*b*c^3*d^3*x^3
+ 10*b*c^6 + (15*b*c^4 - a)*d^2*x^2 - 6*a*c^2 + 4*(5*b*c^5 - a*c)*d*x)/(b*d^4*x
^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a), x)/(b^2*c^8 + 2
*a*b*c^4 + a^2) - 1/((b*c^4 + 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(1/(((d*x + c)^4*b + a)*x^2),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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