3.26.84 \(\int \frac {(-q+p x^2) \sqrt {q^2+p^2 x^4}}{b x^3+a (q+p x^2)^3} \, dx\) [2584]

Optimal. Leaf size=223 \[ \frac {1}{3} \text {RootSum}\left [8 a p^3 q^3+12 a p^2 q^2 \text {$\#$1}^2+8 b \text {$\#$1}^3+6 a p q \text {$\#$1}^4+a \text {$\#$1}^6\& ,\frac {-4 p^2 q^2 \log (x)+4 p^2 q^2 \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right )+4 p q \log (x) \text {$\#$1}^2-4 p q \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-\log (x) \text {$\#$1}^4+\log \left (q+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{4 a p^2 q^2 \text {$\#$1}+4 b \text {$\#$1}^2+4 a p q \text {$\#$1}^3+a \text {$\#$1}^5}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{b x^3+a \left (q+p x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(b*x^3 + a*(q + p*x^2)^3),x]

[Out]

-(q*Defer[Int][Sqrt[q^2 + p^2*x^4]/(b*x^3 + a*(q + p*x^2)^3), x]) + p*Defer[Int][(x^2*Sqrt[q^2 + p^2*x^4])/(b*
x^3 + a*(q + p*x^2)^3), x]

Rubi steps

\begin {align*} \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{b x^3+a \left (q+p x^2\right )^3} \, dx &=\int \left (-\frac {q \sqrt {q^2+p^2 x^4}}{a q^3+3 a p q^2 x^2+b x^3+3 a p^2 q x^4+a p^3 x^6}+\frac {p x^2 \sqrt {q^2+p^2 x^4}}{a q^3+3 a p q^2 x^2+b x^3+3 a p^2 q x^4+a p^3 x^6}\right ) \, dx\\ &=p \int \frac {x^2 \sqrt {q^2+p^2 x^4}}{a q^3+3 a p q^2 x^2+b x^3+3 a p^2 q x^4+a p^3 x^6} \, dx-q \int \frac {\sqrt {q^2+p^2 x^4}}{a q^3+3 a p q^2 x^2+b x^3+3 a p^2 q x^4+a p^3 x^6} \, dx\\ &=p \int \frac {x^2 \sqrt {q^2+p^2 x^4}}{b x^3+a \left (q+p x^2\right )^3} \, dx-q \int \frac {\sqrt {q^2+p^2 x^4}}{b x^3+a \left (q+p x^2\right )^3} \, dx\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(648\) vs. \(2(223)=446\).
time = 3.15, size = 648, normalized size = 2.91 \begin {gather*} \frac {1}{3} \text {RootSum}\left [a+8 b+6 a p q+12 a p^2 q^2+8 a p^3 q^3+6 a \text {$\#$1}+24 b \text {$\#$1}+24 a p q \text {$\#$1}+24 a p^2 q^2 \text {$\#$1}+15 a \text {$\#$1}^2+24 b \text {$\#$1}^2+36 a p q \text {$\#$1}^2+12 a p^2 q^2 \text {$\#$1}^2+20 a \text {$\#$1}^3+8 b \text {$\#$1}^3+24 a p q \text {$\#$1}^3+15 a \text {$\#$1}^4+6 a p q \text {$\#$1}^4+6 a \text {$\#$1}^5+a \text {$\#$1}^6\&,\frac {-\log (x)+4 p q \log (x)-4 p^2 q^2 \log (x)+\log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right )-4 p q \log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right )+4 p^2 q^2 \log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right )-4 \log (x) \text {$\#$1}+8 p q \log (x) \text {$\#$1}+4 \log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}-8 p q \log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}-6 \log (x) \text {$\#$1}^2+4 p q \log (x) \text {$\#$1}^2+6 \log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-4 p q \log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-4 \log (x) \text {$\#$1}^3+4 \log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3-\log (x) \text {$\#$1}^4+\log \left (q-x+p x^2+\sqrt {q^2+p^2 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a+4 b+4 a p q+4 a p^2 q^2+5 a \text {$\#$1}+8 b \text {$\#$1}+12 a p q \text {$\#$1}+4 a p^2 q^2 \text {$\#$1}+10 a \text {$\#$1}^2+4 b \text {$\#$1}^2+12 a p q \text {$\#$1}^2+10 a \text {$\#$1}^3+4 a p q \text {$\#$1}^3+5 a \text {$\#$1}^4+a \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(b*x^3 + a*(q + p*x^2)^3),x]

[Out]

RootSum[a + 8*b + 6*a*p*q + 12*a*p^2*q^2 + 8*a*p^3*q^3 + 6*a*#1 + 24*b*#1 + 24*a*p*q*#1 + 24*a*p^2*q^2*#1 + 15
*a*#1^2 + 24*b*#1^2 + 36*a*p*q*#1^2 + 12*a*p^2*q^2*#1^2 + 20*a*#1^3 + 8*b*#1^3 + 24*a*p*q*#1^3 + 15*a*#1^4 + 6
*a*p*q*#1^4 + 6*a*#1^5 + a*#1^6 & , (-Log[x] + 4*p*q*Log[x] - 4*p^2*q^2*Log[x] + Log[q - x + p*x^2 + Sqrt[q^2
+ p^2*x^4] - x*#1] - 4*p*q*Log[q - x + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1] + 4*p^2*q^2*Log[q - x + p*x^2 + Sqr
t[q^2 + p^2*x^4] - x*#1] - 4*Log[x]*#1 + 8*p*q*Log[x]*#1 + 4*Log[q - x + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#
1 - 8*p*q*Log[q - x + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1 - 6*Log[x]*#1^2 + 4*p*q*Log[x]*#1^2 + 6*Log[q - x
+ p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1^2 - 4*p*q*Log[q - x + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1^2 - 4*L
og[x]*#1^3 + 4*Log[q - x + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1^3 - Log[x]*#1^4 + Log[q - x + p*x^2 + Sqrt[q
^2 + p^2*x^4] - x*#1]*#1^4)/(a + 4*b + 4*a*p*q + 4*a*p^2*q^2 + 5*a*#1 + 8*b*#1 + 12*a*p*q*#1 + 4*a*p^2*q^2*#1
+ 10*a*#1^2 + 4*b*#1^2 + 12*a*p*q*#1^2 + 10*a*#1^3 + 4*a*p*q*#1^3 + 5*a*#1^4 + a*#1^5) & ]/3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.63, size = 722, normalized size = 3.24

method	result	size
default	\(\frac {b \sqrt {p^{2}}\, \left (\munderset {\textit {\_R} =\RootOf \left (p^{4} \textit {\_Z}^{12} a^{2}-12 \sqrt {p^{2}}\, \textit {\_Z}^{11} a^{2} p^{3} q +54 p^{4} \textit {\_Z}^{10} a^{2} q^{2}-\left (100 \sqrt {p^{2}}\, p^{3} a^{2} q^{3}-8 \sqrt {p^{2}}\, b^{2}\right ) \textit {\_Z}^{9}+15 p^{4} \textit {\_Z}^{8} a^{2} q^{4}-\left (-168 \sqrt {p^{2}}\, p^{3} a^{2} q^{5}+24 \sqrt {p^{2}}\, b^{2} q^{2}\right ) \textit {\_Z}^{7}-76 p^{4} \textit {\_Z}^{6} a^{2} q^{6}-\left (168 \sqrt {p^{2}}\, p^{3} a^{2} q^{7}-24 \sqrt {p^{2}}\, b^{2} q^{4}\right ) \textit {\_Z}^{5}+15 p^{4} \textit {\_Z}^{4} a^{2} q^{8}-\left (-100 \sqrt {p^{2}}\, p^{3} a^{2} q^{9}+8 \sqrt {p^{2}}\, b^{2} q^{6}\right ) \textit {\_Z}^{3}+54 p^{4} \textit {\_Z}^{2} a^{2} q^{10}+12 \sqrt {p^{2}}\, \textit {\_Z} \,a^{2} p^{3} q^{11}+a^{2} p^{4} q^{12}\right )}{\sum }\frac {\left (p \left (-\textit {\_R}^{9}+2 q^{4} \textit {\_R}^{5}-q^{8} \textit {\_R} \right )+2 q \left (-\textit {\_R}^{8} \sqrt {p^{2}}-q^{2} \textit {\_R}^{6} \sqrt {p^{2}}+q^{4} \textit {\_R}^{4} \sqrt {p^{2}}+q^{6} \textit {\_R}^{2} \sqrt {p^{2}}\right )\right ) \ln \left (\sqrt {p^{2} x^{4}+q^{2}}-x^{2} \sqrt {p^{2}}-\textit {\_R} \right )}{a^{2} p^{4} \left (-\textit {\_R}^{11}-45 q^{2} \textit {\_R}^{9}-10 q^{4} \textit {\_R}^{7}+38 q^{6} \textit {\_R}^{5}-5 q^{8} \textit {\_R}^{3}-9 q^{10} \textit {\_R} \right )+11 \sqrt {p^{2}}\, \textit {\_R}^{10} a^{2} p^{3} q +75 \sqrt {p^{2}}\, \textit {\_R}^{8} a^{2} p^{3} q^{3}-98 \sqrt {p^{2}}\, \textit {\_R}^{6} a^{2} p^{3} q^{5}+70 \sqrt {p^{2}}\, \textit {\_R}^{4} a^{2} p^{3} q^{7}-25 \sqrt {p^{2}}\, \textit {\_R}^{2} a^{2} p^{3} q^{9}-\sqrt {p^{2}}\, p^{3} a^{2} q^{11}-6 \sqrt {p^{2}}\, \textit {\_R}^{8} b^{2}+14 \sqrt {p^{2}}\, \textit {\_R}^{6} b^{2} q^{2}-10 \sqrt {p^{2}}\, \textit {\_R}^{4} b^{2} q^{4}+2 \sqrt {p^{2}}\, \textit {\_R}^{2} b^{2} q^{6}}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (8 a^{2} \textit {\_Z}^{6}+24 a^{2} p q \,\textit {\_Z}^{4}+24 p^{2} a^{2} q^{2} \textit {\_Z}^{2}+8 a^{2} p^{3} q^{3}-b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\textit {\_R}^{2} p q \right ) \ln \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}+2 p q \,\textit {\_R}^{3}+p^{2} q^{2} \textit {\_R}}\right ) \sqrt {2}}{12 a}\)	\(722\)
elliptic	\(\frac {b \sqrt {p^{2}}\, \left (\munderset {\textit {\_R} =\RootOf \left (p^{4} \textit {\_Z}^{12} a^{2}-12 \sqrt {p^{2}}\, \textit {\_Z}^{11} a^{2} p^{3} q +54 p^{4} \textit {\_Z}^{10} a^{2} q^{2}-\left (100 \sqrt {p^{2}}\, p^{3} a^{2} q^{3}-8 \sqrt {p^{2}}\, b^{2}\right ) \textit {\_Z}^{9}+15 p^{4} \textit {\_Z}^{8} a^{2} q^{4}-\left (-168 \sqrt {p^{2}}\, p^{3} a^{2} q^{5}+24 \sqrt {p^{2}}\, b^{2} q^{2}\right ) \textit {\_Z}^{7}-76 p^{4} \textit {\_Z}^{6} a^{2} q^{6}-\left (168 \sqrt {p^{2}}\, p^{3} a^{2} q^{7}-24 \sqrt {p^{2}}\, b^{2} q^{4}\right ) \textit {\_Z}^{5}+15 p^{4} \textit {\_Z}^{4} a^{2} q^{8}-\left (-100 \sqrt {p^{2}}\, p^{3} a^{2} q^{9}+8 \sqrt {p^{2}}\, b^{2} q^{6}\right ) \textit {\_Z}^{3}+54 p^{4} \textit {\_Z}^{2} a^{2} q^{10}+12 \sqrt {p^{2}}\, \textit {\_Z} \,a^{2} p^{3} q^{11}+a^{2} p^{4} q^{12}\right )}{\sum }\frac {\left (p \left (-\textit {\_R}^{9}+2 q^{4} \textit {\_R}^{5}-q^{8} \textit {\_R} \right )+2 q \left (-\textit {\_R}^{8} \sqrt {p^{2}}-q^{2} \textit {\_R}^{6} \sqrt {p^{2}}+q^{4} \textit {\_R}^{4} \sqrt {p^{2}}+q^{6} \textit {\_R}^{2} \sqrt {p^{2}}\right )\right ) \ln \left (\sqrt {p^{2} x^{4}+q^{2}}-x^{2} \sqrt {p^{2}}-\textit {\_R} \right )}{a^{2} p^{4} \left (-\textit {\_R}^{11}-45 q^{2} \textit {\_R}^{9}-10 q^{4} \textit {\_R}^{7}+38 q^{6} \textit {\_R}^{5}-5 q^{8} \textit {\_R}^{3}-9 q^{10} \textit {\_R} \right )+11 \sqrt {p^{2}}\, \textit {\_R}^{10} a^{2} p^{3} q +75 \sqrt {p^{2}}\, \textit {\_R}^{8} a^{2} p^{3} q^{3}-98 \sqrt {p^{2}}\, \textit {\_R}^{6} a^{2} p^{3} q^{5}+70 \sqrt {p^{2}}\, \textit {\_R}^{4} a^{2} p^{3} q^{7}-25 \sqrt {p^{2}}\, \textit {\_R}^{2} a^{2} p^{3} q^{9}-\sqrt {p^{2}}\, p^{3} a^{2} q^{11}-6 \sqrt {p^{2}}\, \textit {\_R}^{8} b^{2}+14 \sqrt {p^{2}}\, \textit {\_R}^{6} b^{2} q^{2}-10 \sqrt {p^{2}}\, \textit {\_R}^{4} b^{2} q^{4}+2 \sqrt {p^{2}}\, \textit {\_R}^{2} b^{2} q^{6}}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (8 a^{2} \textit {\_Z}^{6}+24 a^{2} p q \,\textit {\_Z}^{4}+24 p^{2} a^{2} q^{2} \textit {\_Z}^{2}+8 a^{2} p^{3} q^{3}-b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\textit {\_R}^{2} p q \right ) \ln \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}+2 p q \,\textit {\_R}^{3}+p^{2} q^{2} \textit {\_R}}\right ) \sqrt {2}}{12 a}\)	\(722\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/(b*x^3+a*(p*x^2+q)^3),x,method=_RETURNVERBOSE)

[Out]

1/6*b*(p^2)^(1/2)*sum((p*(-_R^9+2*_R^5*q^4-_R*q^8)+2*q*(-_R^8*(p^2)^(1/2)-q^2*_R^6*(p^2)^(1/2)+q^4*_R^4*(p^2)^
(1/2)+q^6*_R^2*(p^2)^(1/2)))/(a^2*p^4*(-_R^11-45*_R^9*q^2-10*_R^7*q^4+38*_R^5*q^6-5*_R^3*q^8-9*_R*q^10)+11*(p^
2)^(1/2)*_R^10*a^2*p^3*q+75*(p^2)^(1/2)*_R^8*a^2*p^3*q^3-98*(p^2)^(1/2)*_R^6*a^2*p^3*q^5+70*(p^2)^(1/2)*_R^4*a
^2*p^3*q^7-25*(p^2)^(1/2)*_R^2*a^2*p^3*q^9-(p^2)^(1/2)*p^3*a^2*q^11-6*(p^2)^(1/2)*_R^8*b^2+14*(p^2)^(1/2)*_R^6
*b^2*q^2-10*(p^2)^(1/2)*_R^4*b^2*q^4+2*(p^2)^(1/2)*_R^2*b^2*q^6)*ln((p^2*x^4+q^2)^(1/2)-x^2*(p^2)^(1/2)-_R),_R
=RootOf(p^4*_Z^12*a^2-12*(p^2)^(1/2)*_Z^11*a^2*p^3*q+54*p^4*_Z^10*a^2*q^2-(100*(p^2)^(1/2)*p^3*a^2*q^3-8*(p^2)
^(1/2)*b^2)*_Z^9+15*p^4*_Z^8*a^2*q^4-(-168*(p^2)^(1/2)*p^3*a^2*q^5+24*(p^2)^(1/2)*b^2*q^2)*_Z^7-76*p^4*_Z^6*a^
2*q^6-(168*(p^2)^(1/2)*p^3*a^2*q^7-24*(p^2)^(1/2)*b^2*q^4)*_Z^5+15*p^4*_Z^4*a^2*q^8-(-100*(p^2)^(1/2)*p^3*a^2*
q^9+8*(p^2)^(1/2)*b^2*q^6)*_Z^3+54*p^4*_Z^2*a^2*q^10+12*(p^2)^(1/2)*_Z*a^2*p^3*q^11+a^2*p^4*q^12))+1/12/a*sum(
(_R^4+_R^2*p*q)/(_R^5+2*_R^3*p*q+_R*p^2*q^2)*ln(1/2*(p^2*x^4+q^2)^(1/2)*2^(1/2)/x-_R),_R=RootOf(8*_Z^6*a^2+24*
_Z^4*a^2*p*q+24*_Z^2*a^2*p^2*q^2+8*a^2*p^3*q^3-b^2))*2^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/(b*x^3+a*(p*x^2+q)^3),x, algorithm="maxima")

[Out]

integrate(sqrt(p^2*x^4 + q^2)*(p*x^2 - q)/((p*x^2 + q)^3*a + b*x^3), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/(b*x^3+a*(p*x^2+q)^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (p x^{2} - q\right ) \sqrt {p^{2} x^{4} + q^{2}}}{a p^{3} x^{6} + 3 a p^{2} q x^{4} + 3 a p q^{2} x^{2} + a q^{3} + b x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x**2-q)*(p**2*x**4+q**2)**(1/2)/(b*x**3+a*(p*x**2+q)**3),x)

[Out]

Integral((p*x**2 - q)*sqrt(p**2*x**4 + q**2)/(a*p**3*x**6 + 3*a*p**2*q*x**4 + 3*a*p*q**2*x**2 + a*q**3 + b*x**
3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/(b*x^3+a*(p*x^2+q)^3),x, algorithm="giac")

[Out]

integrate(sqrt(p^2*x^4 + q^2)*(p*x^2 - q)/((p*x^2 + q)^3*a + b*x^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )}{a\,{\left (p\,x^2+q\right )}^3+b\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((p^2*x^4 + q^2)^(1/2)*(q - p*x^2))/(a*(q + p*x^2)^3 + b*x^3),x)

[Out]

int(-((p^2*x^4 + q^2)^(1/2)*(q - p*x^2))/(a*(q + p*x^2)^3 + b*x^3), x)

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