3.30.47 \(\int \frac {(-q+2 p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} (b x^6+a (q+p x^3)^6)}{x^9} \, dx\) [2947]

Optimal. Leaf size=352 \[ \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (6 a q^7-2 a p q^6 x^2+42 a p q^6 x^3-5 a p^2 q^5 x^4-10 a p^2 q^5 x^5+24 b q x^6-15 a p^3 q^4 x^6+126 a p^2 q^5 x^6-15 a p^3 q^4 x^7-20 a p^3 q^4 x^8+24 b p x^9-15 a p^4 q^3 x^9+210 a p^3 q^4 x^9-15 a p^4 q^3 x^{10}-20 a p^4 q^3 x^{11}+210 a p^4 q^3 x^{12}-5 a p^5 q^2 x^{13}-10 a p^5 q^2 x^{14}+126 a p^5 q^2 x^{15}-2 a p^6 q x^{17}+42 a p^6 q x^{18}+6 a p^7 x^{21}\right )}{48 x^8}+\frac {1}{8} \left (8 b p q+5 a p^4 q^4\right ) \log (x)+\frac {1}{8} \left (-8 b p q-5 a p^4 q^4\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \]

[Out]

1/48*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(6*a*p^7*x^21+42*a*p^6*q*x^18-2*a*p^6*q*x^17+126*a*p^5*q^2*x^15-1
0*a*p^5*q^2*x^14-5*a*p^5*q^2*x^13+210*a*p^4*q^3*x^12-20*a*p^4*q^3*x^11-15*a*p^4*q^3*x^10-15*a*p^4*q^3*x^9+210*
a*p^3*q^4*x^9-20*a*p^3*q^4*x^8-15*a*p^3*q^4*x^7-15*a*p^3*q^4*x^6+126*a*p^2*q^5*x^6-10*a*p^2*q^5*x^5-5*a*p^2*q^
5*x^4+42*a*p*q^6*x^3+24*b*p*x^9-2*a*p*q^6*x^2+6*a*q^7+24*b*q*x^6)/x^8+1/8*(5*a*p^4*q^4+8*b*p*q)*ln(x)+1/8*(-5*
a*p^4*q^4-8*b*p*q)*ln(q+p*x^3+(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2))

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Rubi [F]
time = 1.24, 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^6+a \left (q+p x^3\right )^6\right )}{x^9} \, 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^6 + a*(q + p*x^3)^6))/x^9,x]

[Out]

2*p*(b + 5*a*p^2*q^4)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] - a*q^7*Defer[Int][Sqrt[q^2 -
2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x^9, x] - 4*a*p*q^6*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x
^6, x] - q*(b + 3*a*p^2*q^4)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x^3, x] + 25*a*p^4*q^3*Def
er[Int][x^3*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] + 24*a*p^5*q^2*Defer[Int][x^6*Sqrt[q^2 - 2*p*q*x^2
+ 2*p*q*x^3 + p^2*x^6], x] + 11*a*p^6*q*Defer[Int][x^9*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] + 2*a*
p^7*Defer[Int][x^12*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^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx &=\int \left (2 p \left (b+5 a p^2 q^4\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}-\frac {a q^7 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^9}-\frac {4 a p q^6 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^6}-\frac {q \left (b+3 a p^2 q^4\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3}+25 a p^4 q^3 x^3 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+24 a p^5 q^2 x^6 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+11 a p^6 q x^9 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+2 a p^7 x^{12} \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \, dx\\ &=\left (2 a p^7\right ) \int x^{12} \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (11 a p^6 q\right ) \int x^9 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (24 a p^5 q^2\right ) \int x^6 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (25 a p^4 q^3\right ) \int x^3 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-\left (4 a p q^6\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^6} \, dx-\left (a q^7\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^9} \, dx-\left (q \left (b+3 a p^2 q^4\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 (2 p \left (b+5 a p^2 q^4\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 [A]
time = 0.47, size = 210, normalized size = 0.60 \begin {gather*} \frac {1}{48} \left (\frac {\left (q+p x^3\right ) \sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6} \left (24 b x^6+a \left (6 q^6+6 p^6 x^{18}+2 p q^5 x^2 (-1+18 x)+2 p^5 q x^{14} (-1+18 x)+p^2 q^4 x^4 \left (-5-8 x+90 x^2\right )+p^4 q^2 x^{10} \left (-5-8 x+90 x^2\right )+p^3 q^3 x^6 \left (-15-10 x-12 x^2+120 x^3\right )\right )\right )}{x^8}-6 p q \left (8 b+5 a p^3 q^3\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}}{q+p x^3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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^6 + a*(q + p*x^3)^6))/x^9,x]

[Out]

(((q + p*x^3)*Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]*(24*b*x^6 + a*(6*q^6 + 6*p^6*x^18 + 2*p*q^5*x^2*(-1 + 1
8*x) + 2*p^5*q*x^14*(-1 + 18*x) + p^2*q^4*x^4*(-5 - 8*x + 90*x^2) + p^4*q^2*x^10*(-5 - 8*x + 90*x^2) + p^3*q^3
*x^6*(-15 - 10*x - 12*x^2 + 120*x^3))))/x^8 - 6*p*q*(8*b + 5*a*p^3*q^3)*ArcTanh[Sqrt[q^2 + 2*p*q*(-1 + x)*x^2
+ p^2*x^6]/(q + p*x^3)])/48

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2 p \,x^{3}-q \right ) \sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}\, \left (b \,x^{6}+a \left (p \,x^{3}+q \right )^{6}\right )}{x^{9}}\, dx\]

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^6+a*(p*x^3+q)^6)/x^9,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^6+a*(p*x^3+q)^6)/x^9,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)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*x^6+a*(p*x^3+q)^6)/x^9,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)^6*a + b*x^6)*(2*p*x^3 - q)/x^9, 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)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*x^6+a*(p*x^3+q)^6)/x^9,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 (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^{6} x^{18} + 6 a p^{5} q x^{15} + 15 a p^{4} q^{2} x^{12} + 20 a p^{3} q^{3} x^{9} + 15 a p^{2} q^{4} x^{6} + 6 a p q^{5} x^{3} + a q^{6} + b x^{6}\right )}{x^{9}}\, 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**6+a*(p*x**3+q)**6)/x**9,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**6*x**18 + 6*a*p**5*q*x**15 + 15
*a*p**4*q**2*x**12 + 20*a*p**3*q**3*x**9 + 15*a*p**2*q**4*x**6 + 6*a*p*q**5*x**3 + a*q**6 + b*x**6)/x**9, 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((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^6+a*(p*x^3+q)^6)/x^9,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)^6*a + b*x^6)*(2*p*x^3 - q)/x^9, x)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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