3.26.57 \(\int \frac {(-q+2 p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} (b q x+c x^2+b p x^4+a (q+p x^3)^2)}{x^5} \, dx\)

Optimal. Leaf size=215 \[ \frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (3 a p^3 x^9+9 a p^2 q x^6-3 a p^2 q x^5+9 a p q^2 x^3-3 a p q^2 x^2+3 a q^3+4 b p^2 x^7+8 b p q x^4-8 b p q x^3+4 b q^2 x+6 c p x^5+6 c q x^2\right )}{12 x^4}+\frac {1}{2} \left (-a p^2 q^2-2 c p q\right ) \log \left (\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}+p x^3+q\right )+\frac {1}{2} \log (x) \left (a p^2 q^2+2 c p q\right ) \]

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Rubi [F]  time = 1.88, 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+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*q*x + c*x^2 + b*p*x^4 + a*(q + p*x^3)^2))/x
^5,x]

[Out]

2*c*p*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] - a*q^3*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p
*q*x^3 + p^2*x^6]/x^5, x] - b*q^2*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x^4, x] - c*q*Defer[I
nt][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x^3, x] + b*p*q*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 +
p^2*x^6]/x, x] + 3*a*p^2*q*Defer[Int][x*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] + 2*b*p^2*Defer[Int][x
^2*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] + 2*a*p^3*Defer[Int][x^4*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 +
p^2*x^6], x]

Rubi steps

\begin {align*} \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx &=\int \left (2 c p \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}-\frac {a q^3 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^5}-\frac {b q^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^4}-\frac {c q \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3}+\frac {b p q \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x}+3 a p^2 q x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+2 b p^2 x^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+2 a p^3 x^4 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \, dx\\ &=(2 c p) \int \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (2 b p^2\right ) \int x^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (2 a p^3\right ) \int x^4 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-(c q) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3} \, dx+(b p q) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x} \, dx+\left (3 a p^2 q\right ) \int x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-\left (b q^2\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^4} \, dx-\left (a q^3\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 1.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*q*x + c*x^2 + b*p*x^4 + a*(q + p*x^3)
^2))/x^5,x]

[Out]

Integrate[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*q*x + c*x^2 + b*p*x^4 + a*(q + p*x^3)
^2))/x^5, x]

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IntegrateAlgebraic [A]  time = 0.64, size = 215, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (3 a q^3+4 b q^2 x+6 c q x^2-3 a p q^2 x^2-8 b p q x^3+9 a p q^2 x^3+8 b p q x^4+6 c p x^5-3 a p^2 q x^5+9 a p^2 q x^6+4 b p^2 x^7+3 a p^3 x^9\right )}{12 x^4}+\frac {1}{2} \left (2 c p q+a p^2 q^2\right ) \log (x)+\frac {1}{2} \left (-2 c p q-a p^2 q^2\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*q*x + c*x^2 + b*p*x^4 + a*(q
+ p*x^3)^2))/x^5,x]

[Out]

(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(3*a*q^3 + 4*b*q^2*x + 6*c*q*x^2 - 3*a*p*q^2*x^2 - 8*b*p*q*x^3 +
9*a*p*q^2*x^3 + 8*b*p*q*x^4 + 6*c*p*x^5 - 3*a*p^2*q*x^5 + 9*a*p^2*q*x^6 + 4*b*p^2*x^7 + 3*a*p^3*x^9))/(12*x^4)
+ ((2*c*p*q + a*p^2*q^2)*Log[x])/2 + ((-2*c*p*q - a*p^2*q^2)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3
+ p^2*x^6]])/2

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fricas [F(-1)]  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((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*q*x+c*x^2+b*p*x^4+a*(p*x^3+q)^2)/x^5,x, algor
ithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left (b p x^{4} + {\left (p x^{3} + q\right )}^{2} a + b q x + c x^{2}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*q*x+c*x^2+b*p*x^4+a*(p*x^3+q)^2)/x^5,x, algor
ithm="giac")

[Out]

integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(b*p*x^4 + (p*x^3 + q)^2*a + b*q*x + c*x^2)*(2*p*x^3 - q
)/x^5, x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (2 p \,x^{3}-q \right ) \sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}\, \left (b q x +c \,x^{2}+b p \,x^{4}+a \left (p \,x^{3}+q \right )^{2}\right )}{x^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*q*x+c*x^2+b*p*x^4+a*(p*x^3+q)^2)/x^5,x)

[Out]

int((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*q*x+c*x^2+b*p*x^4+a*(p*x^3+q)^2)/x^5,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left (b p x^{4} + {\left (p x^{3} + q\right )}^{2} a + b q x + c x^{2}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*q*x+c*x^2+b*p*x^4+a*(p*x^3+q)^2)/x^5,x, algor
ithm="maxima")

[Out]

integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(b*p*x^4 + (p*x^3 + q)^2*a + b*q*x + c*x^2)*(2*p*x^3 - q
)/x^5, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (q-2\,p\,x^3\right )\,\left (a\,{\left (p\,x^3+q\right )}^2+c\,x^2+b\,p\,x^4+b\,q\,x\right )\,\sqrt {p^2\,x^6+2\,p\,q\,x^3-2\,p\,q\,x^2+q^2}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((q - 2*p*x^3)*(a*(q + p*x^3)^2 + c*x^2 + b*p*x^4 + b*q*x)*(p^2*x^6 + q^2 - 2*p*q*x^2 + 2*p*q*x^3)^(1/2))
/x^5,x)

[Out]

-int(((q - 2*p*x^3)*(a*(q + p*x^3)^2 + c*x^2 + b*p*x^4 + b*q*x)*(p^2*x^6 + q^2 - 2*p*q*x^2 + 2*p*q*x^3)^(1/2))
/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 p x^{3} - q\right ) \sqrt {p^{2} x^{6} + 2 p q x^{3} - 2 p q x^{2} + q^{2}} \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b p x^{4} + b q x + c x^{2}\right )}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*p*x**3-q)*(p**2*x**6+2*p*q*x**3-2*p*q*x**2+q**2)**(1/2)*(b*q*x+c*x**2+b*p*x**4+a*(p*x**3+q)**2)/x
**5,x)

[Out]

Integral((2*p*x**3 - q)*sqrt(p**2*x**6 + 2*p*q*x**3 - 2*p*q*x**2 + q**2)*(a*p**2*x**6 + 2*a*p*q*x**3 + a*q**2
+ b*p*x**4 + b*q*x + c*x**2)/x**5, x)

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