3.17.88 \(\int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} (b^6+a d-(6 b^5+d) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6)} \, dx\)

Optimal. Leaf size=113 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{(b-x)^2}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{(b-x)^2}\right )}{d^{3/4}} \]

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Rubi [F]  time = 28.01, 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 {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-b + x)*(-6*a + b + 5*x))/(((-a + x)*(-b + x)^2)^(1/4)*(b^6 + a*d - (6*b^5 + d)*x + 15*b^4*x^2 - 20*b^3*
x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6)),x]

[Out]

(20*a*(-a + x)^(1/4)*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^2*Sqrt[a - b + x^4])/(a^6*(1 + b^6/a^6) - 6*b^5*(
1 + d/(6*b^5))*x^4 + 15*b^4*x^8 - 20*b^3*x^12 + 15*b^2*x^16 - 6*b*x^20 + x^24 - 6*a^5*(b - x^4) + 15*a^4*(b -
x^4)^2 - 20*a^3*(b - x^4)^3 + 15*a^2*(b - x^4)^4 - 6*a*(b - x^4)^5), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x
)^2))^(1/4) + (20*(-a + x)^(1/4)*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^6*Sqrt[a - b + x^4])/(a^6*(1 + b^6/a^
6) - 6*b^5*(1 + d/(6*b^5))*x^4 + 15*b^4*x^8 - 20*b^3*x^12 + 15*b^2*x^16 - 6*b*x^20 + x^24 - 6*a^5*(b - x^4) +
15*a^4*(b - x^4)^2 - 20*a^3*(b - x^4)^3 + 15*a^2*(b - x^4)^4 - 6*a*(b - x^4)^5), x], x, (-a + x)^(1/4)])/(-((a
- x)*(b - x)^2))^(1/4) - (4*(6*a - b)*(-a + x)^(1/4)*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^2*Sqrt[a - b + x
^4])/(b^6*(1 + (a*d)/b^6) - (6*b^5 + d)*(a + x^4) + 15*b^4*(a + x^4)^2 - 20*b^3*(a + x^4)^3 + 15*b^2*(a + x^4)
^4 - 6*b*(a + x^4)^5 + (a + x^4)^6), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(1/4)

Rubi steps

\begin {align*} \int \frac {(-b+x) (-6 a+b+5 x)}{\sqrt [4]{(-a+x) (-b+x)^2} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x} (-6 a+b+5 x)}{\sqrt [4]{-a+x} \left (b^6+a d-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x} (-6 a+b+5 x)}{\sqrt [4]{-a+x} \left (b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {6 a \left (1-\frac {b}{6 a}\right ) \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (-b^6 \left (1+\frac {a d}{b^6}\right )+6 b^5 \left (1+\frac {d}{6 b^5}\right ) x-15 b^4 x^2+20 b^3 x^3-15 b^2 x^4+6 b x^5-x^6\right )}+\frac {5 x \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (b^6 \left (1+\frac {a d}{b^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (5 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {x \sqrt {-b+x}}{\sqrt [4]{-a+x} \left (b^6 \left (1+\frac {a d}{b^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left ((6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x}}{\sqrt [4]{-a+x} \left (-b^6 \left (1+\frac {a d}{b^6}\right )+6 b^5 \left (1+\frac {d}{6 b^5}\right ) x-15 b^4 x^2+20 b^3 x^3-15 b^2 x^4+6 b x^5-x^6\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^4\right ) \sqrt {a-b+x^4}}{b^6+a d-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6+a d-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+x^4\right ) \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^2 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}}+\frac {x^6 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (20 a \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b \left (-6 a^5+15 a^4 b-20 a^3 b^2+15 a^2 b^3-6 a b^4+b^5\right )}{a^6}\right )+6 a^5 \left (1-\frac {30 a^4 b-60 a^3 b^2+60 a^2 b^3-30 a b^4+6 b^5+d}{6 a^5}\right ) x^4+15 a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^{16}+6 a \left (1-\frac {b}{a}\right ) x^{20}+x^{24}} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (20 \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b^6}{a^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x^4+15 b^4 x^8-20 b^3 x^{12}+15 b^2 x^{16}-6 b x^{20}+x^{24}-6 a^5 \left (b-x^4\right )+15 a^4 \left (b-x^4\right )^2-20 a^3 \left (b-x^4\right )^3+15 a^2 \left (b-x^4\right )^4-6 a \left (b-x^4\right )^5} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}+\frac {\left (20 a \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{a^6 \left (1+\frac {b^6}{a^6}\right )-6 b^5 \left (1+\frac {d}{6 b^5}\right ) x^4+15 b^4 x^8-20 b^3 x^{12}+15 b^2 x^{16}-6 b x^{20}+x^{24}-6 a^5 \left (b-x^4\right )+15 a^4 \left (b-x^4\right )^2-20 a^3 \left (b-x^4\right )^3+15 a^2 \left (b-x^4\right )^4-6 a \left (b-x^4\right )^5} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}-\frac {\left (4 (6 a-b) \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a-b+x^4}}{b^6 \left (1+\frac {a d}{b^6}\right )-\left (6 b^5+d\right ) \left (a+x^4\right )+15 b^4 \left (a+x^4\right )^2-20 b^3 \left (a+x^4\right )^3+15 b^2 \left (a+x^4\right )^4-6 b \left (a+x^4\right )^5+\left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x)^2}}\\ \end {align*}

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Mathematica [C]  time = 22.59, size = 333050, normalized size = 2947.35 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-b + x)*(-6*a + b + 5*x))/(((-a + x)*(-b + x)^2)^(1/4)*(b^6 + a*d - (6*b^5 + d)*x + 15*b^4*x^2 - 2
0*b^3*x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6)),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.45, size = 113, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-b + x)*(-6*a + b + 5*x))/(((-a + x)*(-b + x)^2)^(1/4)*(b^6 + a*d - (6*b^5 + d)*x + 15*b^
4*x^2 - 20*b^3*x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6)),x]

[Out]

(2*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)^2])/d^(3/4) - (2*ArcTanh
[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)^2])/d^(3/4)

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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((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x+15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*
b*x^5+x^6),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 (6 \, a - b - 5 \, x\right )} {\left (b - x\right )}}{{\left (b^{6} + 15 \, b^{4} x^{2} - 20 \, b^{3} x^{3} + 15 \, b^{2} x^{4} - 6 \, b x^{5} + x^{6} + a d - {\left (6 \, b^{5} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x+15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*
b*x^5+x^6),x, algorithm="giac")

[Out]

integrate((6*a - b - 5*x)*(b - x)/((b^6 + 15*b^4*x^2 - 20*b^3*x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6 + a*d - (6*b^5
+ d)*x)*(-(a - x)*(b - x)^2)^(1/4)), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (-6 a +b +5 x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (b^{6}+a d -\left (6 b^{5}+d \right ) x +15 b^{4} x^{2}-20 b^{3} x^{3}+15 b^{2} x^{4}-6 b \,x^{5}+x^{6}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x+15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*b*x^5+
x^6),x)

[Out]

int((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x+15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*b*x^5+
x^6),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (6 \, a - b - 5 \, x\right )} {\left (b - x\right )}}{{\left (b^{6} + 15 \, b^{4} x^{2} - 20 \, b^{3} x^{3} + 15 \, b^{2} x^{4} - 6 \, b x^{5} + x^{6} + a d - {\left (6 \, b^{5} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)^2)^(1/4)/(b^6+a*d-(6*b^5+d)*x+15*b^4*x^2-20*b^3*x^3+15*b^2*x^4-6*
b*x^5+x^6),x, algorithm="maxima")

[Out]

integrate((6*a - b - 5*x)*(b - x)/((b^6 + 15*b^4*x^2 - 20*b^3*x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6 + a*d - (6*b^5
+ d)*x)*(-(a - x)*(b - x)^2)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (b-x\right )\,\left (b-6\,a+5\,x\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/4}\,\left (a\,d-6\,b\,x^5-x\,\left (6\,b^5+d\right )+b^6+x^6+15\,b^2\,x^4-20\,b^3\,x^3+15\,b^4\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b - x)*(b - 6*a + 5*x))/((-(a - x)*(b - x)^2)^(1/4)*(a*d - 6*b*x^5 - x*(d + 6*b^5) + b^6 + x^6 + 15*b^2
*x^4 - 20*b^3*x^3 + 15*b^4*x^2)),x)

[Out]

int(-((b - x)*(b - 6*a + 5*x))/((-(a - x)*(b - x)^2)^(1/4)*(a*d - 6*b*x^5 - x*(d + 6*b^5) + b^6 + x^6 + 15*b^2
*x^4 - 20*b^3*x^3 + 15*b^4*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- b + x\right ) \left (- 6 a + b + 5 x\right )}{\sqrt [4]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (a d + b^{6} - 6 b^{5} x + 15 b^{4} x^{2} - 20 b^{3} x^{3} + 15 b^{2} x^{4} - 6 b x^{5} - d x + x^{6}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b+x)*(-6*a+b+5*x)/((-a+x)*(-b+x)**2)**(1/4)/(b**6+a*d-(6*b**5+d)*x+15*b**4*x**2-20*b**3*x**3+15*b*
*2*x**4-6*b*x**5+x**6),x)

[Out]

Integral((-b + x)*(-6*a + b + 5*x)/(((-a + x)*(-b + x)**2)**(1/4)*(a*d + b**6 - 6*b**5*x + 15*b**4*x**2 - 20*b
**3*x**3 + 15*b**2*x**4 - 6*b*x**5 - d*x + x**6)), x)

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