3.27.78 \(\int \frac {(-q+2 p x^3) (a q+b x+a p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3 (c q+d x+c p x^3)} \, dx\) [2678]

Optimal. Leaf size=242 \[ \frac {\left (a c q+2 b c x-2 a d x+a c p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{2 c^2 x^2}-\frac {2 (-b c+a d) \sqrt {-d^2+2 c^2 p q} \text {ArcTan}\left (\frac {\sqrt {-d^2+2 c^2 p q} x}{c q+d x+c p x^3+c \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right )}{c^3}+\frac {\left (b c d-a d^2+a c^2 p q\right ) \log (x)}{c^3}+\frac {\left (-b c d+a d^2-a c^2 p q\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right )}{c^3} \]

[Out]

1/2*(a*c*p*x^3+a*c*q-2*a*d*x+2*b*c*x)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)/c^2/x^2-2*(a*d-b*c)*(2*c^2*p*q-d
^2)^(1/2)*arctan((2*c^2*p*q-d^2)^(1/2)*x/(c*q+d*x+c*p*x^3+c*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)))/c^3+(a*c
^2*p*q-a*d^2+b*c*d)*ln(x)/c^3+(-a*c^2*p*q+a*d^2-b*c*d)*ln(q+p*x^3+(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2))/c^3

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Rubi [F]
time = 13.29, 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 ) \left (a q+b x+a p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3 \left (c q+d x+c p x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*a*p*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x])/c - (a*q*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 +
2*p*q*x^3 + p^2*x^6]/x^3, x])/c - ((b*c - a*d)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x^2, x])
/c^2 + (d*(b*c - a*d)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x, x])/(c^3*q) - (d^2*(b*c - a*d)
*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/(c*q + d*x + c*p*x^3), x])/(c^3*q) + (3*(b*c - a*d)*p*
Defer[Int][(x*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(c*q + d*x + c*p*x^3), x])/c - (d*(b*c - a*d)*p*Def
er[Int][(x^2*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(c*q + d*x + c*p*x^3), x])/(c^2*q)

Rubi steps

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

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Mathematica [A]
time = 1.11, size = 206, normalized size = 0.85 \begin {gather*} \frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6} \left (2 b c x-2 a d x+a c \left (q+p x^3\right )\right )}{2 c^2 x^2}-\frac {2 (-b c+a d) \sqrt {-d^2+2 c^2 p q} \text {ArcTan}\left (\frac {\sqrt {-d^2+2 c^2 p q} x}{d x+c \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )}\right )}{c^3}-\frac {\left (b c d-a d^2+a c^2 p q\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}}{q+p x^3}\right )}{c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]*(2*b*c*x - 2*a*d*x + a*c*(q + p*x^3)))/(2*c^2*x^2) - (2*(-(b*c) + a*
d)*Sqrt[-d^2 + 2*c^2*p*q]*ArcTan[(Sqrt[-d^2 + 2*c^2*p*q]*x)/(d*x + c*(q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x^
2 + p^2*x^6]))])/c^3 - ((b*c*d - a*d^2 + a*c^2*p*q)*ArcTanh[Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]/(q + p*x^
3)])/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(a*p*x^3 + a*q + b*x)*(2*p*x^3 - q)/((c*p*x^3 + c*q + d*
x)*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((2*p*x^3-q)*(a*p*x^3+a*q+b*x)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)/x^3/(c*p*x^3+c*q+d*x),x, algor
ithm="fricas")

[Out]

Timed out

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Sympy [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((2*p*x**3-q)*(a*p*x**3+a*q+b*x)*(p**2*x**6+2*p*q*x**3-2*p*q*x**2+q**2)**(1/2)/x**3/(c*p*x**3+c*q+d*x
),x)

[Out]

Timed out

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

[Out]

integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(a*p*x^3 + a*q + b*x)*(2*p*x^3 - q)/((c*p*x^3 + c*q + d*
x)*x^3), 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\,p\,x^3+b\,x+a\,q\right )\,\sqrt {p^2\,x^6+2\,p\,q\,x^3-2\,p\,q\,x^2+q^2}}{x^3\,\left (c\,p\,x^3+d\,x+c\,q\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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