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

Optimal. Leaf size=191 \[ \frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (6 a p^4 x^{12}+24 a p^3 q x^9-4 a p^3 q x^8+36 a p^2 q^2 x^6-8 a p^2 q^2 x^5-16 a p^2 q^2 x^4+24 a p q^3 x^3-4 a p q^3 x^2+6 a q^4+15 b p x^6+15 b q x^3\right )}{30 x^5}-b p q \log \left (\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}+p x^3+q\right )+b p q \log (x) \]

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Rubi [F]  time = 1.32, 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 x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, 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*x^3 + a*(q + p*x^3)^3))/x^6,x]

[Out]

p*(2*b + 3*a*p*q^2)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] - a*q^4*Defer[Int][Sqrt[q^2 - 2
*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x^6, x] - q*(b + a*p*q^2)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^
6]/x^3, x] + 5*a*p^3*q*Defer[Int][x^3*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] + 2*a*p^4*Defer[Int][x^6
*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 x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, dx &=\int \left (p \left (2 b+3 a p q^2\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}-\frac {a q^4 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^6}-\frac {q \left (b+a p q^2\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3}+5 a p^3 q x^3 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+2 a p^4 x^6 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \, dx\\ &=\left (2 a p^4\right ) \int x^6 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (5 a p^3 q\right ) \int x^3 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-\left (a q^4\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^6} \, dx-\left (q \left (b+a p q^2\right )\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3} \, dx+\left (p \left (2 b+3 a p q^2\right )\right ) \int \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.62, 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 x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, 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*x^3 + a*(q + p*x^3)^3))/x^6,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*x^3 + a*(q + p*x^3)^3))/x^6, x]

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IntegrateAlgebraic [A]  time = 0.49, size = 191, 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 (6 a q^4-4 a p q^3 x^2+15 b q x^3+24 a p q^3 x^3-16 a p^2 q^2 x^4-8 a p^2 q^2 x^5+15 b p x^6+36 a p^2 q^2 x^6-4 a p^3 q x^8+24 a p^3 q x^9+6 a p^4 x^{12}\right )}{30 x^5}+b p q \log (x)-b p q \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*x^3 + a*(q + p*x^3)^3))/x^6,
x]

[Out]

(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(6*a*q^4 - 4*a*p*q^3*x^2 + 15*b*q*x^3 + 24*a*p*q^3*x^3 - 16*a*p^2
*q^2*x^4 - 8*a*p^2*q^2*x^5 + 15*b*p*x^6 + 36*a*p^2*q^2*x^6 - 4*a*p^3*q*x^8 + 24*a*p^3*q*x^9 + 6*a*p^4*x^12))/(
30*x^5) + b*p*q*Log[x] - b*p*q*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]]

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fricas [A]  time = 40.38, size = 188, normalized size = 0.98 \begin {gather*} -\frac {30 \, b p q x^{5} \log \left (\frac {p x^{3} + q + \sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}}}{x}\right ) - {\left (6 \, a p^{4} x^{12} + 24 \, a p^{3} q x^{9} - 4 \, a p^{3} q x^{8} - 8 \, a p^{2} q^{2} x^{5} - 16 \, a p^{2} q^{2} x^{4} - 4 \, a p q^{3} x^{2} + 3 \, {\left (12 \, a p^{2} q^{2} + 5 \, b p\right )} x^{6} + 6 \, a q^{4} + 3 \, {\left (8 \, a p q^{3} + 5 \, b q\right )} x^{3}\right )} \sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}}}{30 \, x^{5}} \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*x^3+a*(p*x^3+q)^3)/x^6,x, algorithm="fricas")

[Out]

-1/30*(30*b*p*q*x^5*log((p*x^3 + q + sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2))/x) - (6*a*p^4*x^12 + 24*a*p^
3*q*x^9 - 4*a*p^3*q*x^8 - 8*a*p^2*q^2*x^5 - 16*a*p^2*q^2*x^4 - 4*a*p*q^3*x^2 + 3*(12*a*p^2*q^2 + 5*b*p)*x^6 +
6*a*q^4 + 3*(8*a*p*q^3 + 5*b*q)*x^3)*sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2))/x^5

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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 ({\left (p x^{3} + q\right )}^{3} a + b x^{3}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{6}}\,{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*x^3+a*(p*x^3+q)^3)/x^6,x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2 p \,x^{3}-q \right ) \sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}\, \left (b \,x^{3}+a \left (p \,x^{3}+q \right )^{3}\right )}{x^{6}}\, dx\]

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*x^3+a*(p*x^3+q)^3)/x^6,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*x^3+a*(p*x^3+q)^3)/x^6,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 ({\left (p x^{3} + q\right )}^{3} a + b x^{3}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{6}}\,{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*x^3+a*(p*x^3+q)^3)/x^6,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (q-2\,p\,x^3\right )\,\left (a\,{\left (p\,x^3+q\right )}^3+b\,x^3\right )\,\sqrt {p^2\,x^6+2\,p\,q\,x^3-2\,p\,q\,x^2+q^2}}{x^6} \,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)^3 + b*x^3)*(p^2*x^6 + q^2 - 2*p*q*x^2 + 2*p*q*x^3)^(1/2))/x^6,x)

[Out]

-int(((q - 2*p*x^3)*(a*(q + p*x^3)^3 + b*x^3)*(p^2*x^6 + q^2 - 2*p*q*x^2 + 2*p*q*x^3)^(1/2))/x^6, 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^{3} x^{9} + 3 a p^{2} q x^{6} + 3 a p q^{2} x^{3} + a q^{3} + b x^{3}\right )}{x^{6}}\, 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*x**3+a*(p*x**3+q)**3)/x**6,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**3*x**9 + 3*a*p**2*q*x**6 + 3*a*
p*q**2*x**3 + a*q**3 + b*x**3)/x**6, x)

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