3.30.48 \(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} (b x^{12}+a (q+p x^3)^6)}{x^{17}} \, dx\)

Optimal. Leaf size=352 \[ \frac {1}{4} \log (x) \left (5 a p^4 q^4+8 b p q\right )+\frac {1}{8} \left (-5 a p^4 q^4-8 b p q\right ) \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+\frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (6 a p^7 x^{21}-2 a p^6 q x^{19}+42 a p^6 q x^{18}-5 a p^5 q^2 x^{17}-10 a p^5 q^2 x^{16}+126 a p^5 q^2 x^{15}-15 a p^4 q^3 x^{15}-15 a p^4 q^3 x^{14}-20 a p^4 q^3 x^{13}+210 a p^4 q^3 x^{12}-15 a p^3 q^4 x^{12}-15 a p^3 q^4 x^{11}-20 a p^3 q^4 x^{10}+210 a p^3 q^4 x^9-5 a p^2 q^5 x^8-10 a p^2 q^5 x^7+126 a p^2 q^5 x^6-2 a p q^6 x^4+42 a p q^6 x^3+6 a q^7+24 b p x^{15}+24 b q x^{12}\right )}{48 x^{16}} \]

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Rubi [F]  time = 1.82, 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 (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^12 + a*(q + p*x^3)^6))/x^17,x]

[Out]

-2*a*q^7*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^17, x] - 11*a*p*q^6*Defer[Int][Sqrt[q^2 + 2*
p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^14, x] - 24*a*p^2*q^5*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/
x^11, x] - 25*a*p^3*q^4*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^8, x] - 2*q*(b + 5*a*p^4*q^2)
*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^5, x] + p*(b + 3*a*p^4*q^2)*Defer[Int][Sqrt[q^2 + 2*
p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^2, x] + 4*a*p^6*q*Defer[Int][x*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6], x
] + a*p^7*Defer[Int][x^4*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6], x]

Rubi steps

\begin {align*} \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx &=\int \left (-\frac {2 a q^7 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{17}}-\frac {11 a p q^6 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{14}}-\frac {24 a p^2 q^5 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}}-\frac {25 a p^3 q^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8}-\frac {2 q \left (b+5 a p^4 q^2\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}+\frac {p \left (b+3 a p^4 q^2\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}+4 a p^6 q x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}+a p^7 x^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \, dx\\ &=\left (a p^7\right ) \int x^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx+\left (4 a p^6 q\right ) \int x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx-\left (25 a p^3 q^4\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8} \, dx-\left (24 a p^2 q^5\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}} \, dx-\left (11 a p q^6\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{14}} \, dx-\left (2 a q^7\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{17}} \, dx+\left (p \left (b+3 a p^4 q^2\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx-\left (2 q \left (b+5 a p^4 q^2\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 1.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^12 + a*(q + p*x^3)^6))/x^17,x]

[Out]

Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^12 + a*(q + p*x^3)^6))/x^17, x]

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IntegrateAlgebraic [A]  time = 0.58, size = 352, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (6 a q^7+42 a p q^6 x^3-2 a p q^6 x^4+126 a p^2 q^5 x^6-10 a p^2 q^5 x^7-5 a p^2 q^5 x^8+210 a p^3 q^4 x^9-20 a p^3 q^4 x^{10}-15 a p^3 q^4 x^{11}+24 b q x^{12}+210 a p^4 q^3 x^{12}-15 a p^3 q^4 x^{12}-20 a p^4 q^3 x^{13}-15 a p^4 q^3 x^{14}+24 b p x^{15}+126 a p^5 q^2 x^{15}-15 a p^4 q^3 x^{15}-10 a p^5 q^2 x^{16}-5 a p^5 q^2 x^{17}+42 a p^6 q x^{18}-2 a p^6 q x^{19}+6 a p^7 x^{21}\right )}{48 x^{16}}+\frac {1}{4} \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^3-2 p q x^4+p^2 x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^12 + a*(q + p*x^3)^6))/x^1
7,x]

[Out]

(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(6*a*q^7 + 42*a*p*q^6*x^3 - 2*a*p*q^6*x^4 + 126*a*p^2*q^5*x^6 - 1
0*a*p^2*q^5*x^7 - 5*a*p^2*q^5*x^8 + 210*a*p^3*q^4*x^9 - 20*a*p^3*q^4*x^10 - 15*a*p^3*q^4*x^11 + 24*b*q*x^12 +
210*a*p^4*q^3*x^12 - 15*a*p^3*q^4*x^12 - 20*a*p^4*q^3*x^13 - 15*a*p^4*q^3*x^14 + 24*b*p*x^15 + 126*a*p^5*q^2*x
^15 - 15*a*p^4*q^3*x^15 - 10*a*p^5*q^2*x^16 - 5*a*p^5*q^2*x^17 + 42*a*p^6*q*x^18 - 2*a*p^6*q*x^19 + 6*a*p^7*x^
21))/(48*x^16) + ((8*b*p*q + 5*a*p^4*q^4)*Log[x])/4 + ((-8*b*p*q - 5*a*p^4*q^4)*Log[q + p*x^3 + Sqrt[q^2 + 2*p
*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/8

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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((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p*x^3+q)^6)/x^17,x, algorithm="fricas
")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{12} + {\left (p x^{3} + q\right )}^{6} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{17}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p*x^3+q)^6)/x^17,x, algorithm="giac")

[Out]

integrate((b*x^12 + (p*x^3 + q)^6*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^17, x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 x^{4} p q +2 p q \,x^{3}+q^{2}}\, \left (b \,x^{12}+a \left (p \,x^{3}+q \right )^{6}\right )}{x^{17}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p*x^3+q)^6)/x^17,x)

[Out]

int((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p*x^3+q)^6)/x^17,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{12} + {\left (p x^{3} + q\right )}^{6} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{17}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p*x^3+q)^6)/x^17,x, algorithm="maxima
")

[Out]

integrate((b*x^12 + (p*x^3 + q)^6*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^17, 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(-((a*(q + p*x^3)^6 + b*x^12)*(2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2))/x^17,x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p^{2} x^{6} - 2 p q x^{4} + 2 p q x^{3} + 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^{12}\right )}{x^{17}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x**3-2*q)*(p**2*x**6-2*p*q*x**4+2*p*q*x**3+q**2)**(1/2)*(b*x**12+a*(p*x**3+q)**6)/x**17,x)

[Out]

Integral((p*x**3 - 2*q)*sqrt(p**2*x**6 - 2*p*q*x**4 + 2*p*q*x**3 + 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**12)/x**17, x)

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