3.17.89 \(\int \frac {(-6 a+b+5 x) (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5)}{((-a+x) (-b+x)^2)^{3/4} (a+b^6 d-(1+6 b^5 d) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6)} \, dx\)

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

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Rubi [F]  time = 52.45, 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 {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-6*a + b + 5*x)*(-b^5 + 5*b^4*x - 10*b^3*x^2 + 10*b^2*x^3 - 5*b*x^4 + x^5))/(((-a + x)*(-b + x)^2)^(3/4)
*(a + b^6*d - (1 + 6*b^5*d)*x + 15*b^4*d*x^2 - 20*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*x^6)),x]

[Out]

(20*a*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(a - b + x^4)^(7/2)/(a^6*(1 + (b*(-6*a^5 + 15*a^4*
b - 20*a^3*b^2 + 15*a^2*b^3 - 6*a*b^4 + b^5))/a^6)*d - (1 - 6*(a - b)^5*d)*x^4 + 15*a^4*(1 + (b*(-4*a^3 + 6*a^
2*b - 4*a*b^2 + b^3))/a^4)*d*x^8 + 20*a^3*(1 - (b*(3*a^2 - 3*a*b + b^2))/a^3)*d*x^12 + 15*a^2*(1 + (b*(-2*a +
b))/a^2)*d*x^16 + 6*a*(1 - b/a)*d*x^20 + d*x^24), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(3/4) + (20*(
-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(x^4*(a - b + x^4)^(7/2))/(a^6*(1 + (b*(-6*a^5 + 15*a^4*b
- 20*a^3*b^2 + 15*a^2*b^3 - 6*a*b^4 + b^5))/a^6)*d - (1 - 6*(a - b)^5*d)*x^4 + 15*a^4*(1 + (b*(-4*a^3 + 6*a^2
*b - 4*a*b^2 + b^3))/a^4)*d*x^8 + 20*a^3*(1 - (b*(3*a^2 - 3*a*b + b^2))/a^3)*d*x^12 + 15*a^2*(1 + (b*(-2*a + b
))/a^2)*d*x^16 + 6*a*(1 - b/a)*d*x^20 + d*x^24), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(3/4) - (4*(6*
a - b)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(a - b + x^4)^(7/2)/(a*(1 + (b^6*d)/a) - (1 + 6*b
^5*d)*(a + x^4) + 15*b^4*d*(a + x^4)^2 - 20*b^3*d*(a + x^4)^3 + 15*b^2*d*(a + x^4)^4 - 6*b*d*(a + x^4)^5 + d*(
a + x^4)^6), x], x, (-a + x)^(1/4)])/(-((a - x)*(b - x)^2))^(3/4)

Rubi steps

\begin {align*} \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{(-a+x)^{3/4} (-b+x)^{3/2} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {(-6 a+b+5 x) \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4\right )}{(-a+x)^{3/4} \sqrt {-b+x} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt {-b+x} (-6 a+b+5 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{(-a+x)^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {(6 a-b-5 x) \sqrt {-b+x} \left (b^3-3 b^2 x+3 b x^2-x^3\right )}{(-a+x)^{3/4} \left (a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {(-b+x)^{7/2} (-6 a+b+5 x)}{(-a+x)^{3/4} \left (a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left ((-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \left (\frac {6 a \left (1-\frac {b}{6 a}\right ) (-b+x)^{7/2}}{(-a+x)^{3/4} \left (-a \left (1+\frac {b^6 d}{a}\right )+\left (1+6 b^5 d\right ) x-15 b^4 d x^2+20 b^3 d x^3-15 b^2 d x^4+6 b d x^5-d x^6\right )}+\frac {5 x (-b+x)^{7/2}}{(-a+x)^{3/4} \left (a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (5 (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {x (-b+x)^{7/2}}{(-a+x)^{3/4} \left (a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left ((6 a-b) (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {(-b+x)^{7/2}}{(-a+x)^{3/4} \left (-a \left (1+\frac {b^6 d}{a}\right )+\left (1+6 b^5 d\right ) x-15 b^4 d x^2+20 b^3 d x^3-15 b^2 d x^4+6 b d x^5-d x^6\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (20 (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (a+x^4\right ) \left (a-b+x^4\right )^{7/2}}{a+b^6 d-\left (1+6 b^5 d\right ) \left (a+x^4\right )+15 b^4 d \left (a+x^4\right )^2-20 b^3 d \left (a+x^4\right )^3+15 b^2 d \left (a+x^4\right )^4-6 b d \left (a+x^4\right )^5+d \left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (4 (6 a-b) (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^4\right )^{7/2}}{a+b^6 d-\left (1+6 b^5 d\right ) \left (a+x^4\right )+15 b^4 d \left (a+x^4\right )^2-20 b^3 d \left (a+x^4\right )^3+15 b^2 d \left (a+x^4\right )^4-6 b d \left (a+x^4\right )^5+d \left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (20 (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (a+x^4\right ) \left (a-b+x^4\right )^{7/2}}{a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) \left (a+x^4\right )+15 b^4 d \left (a+x^4\right )^2-20 b^3 d \left (a+x^4\right )^3+15 b^2 d \left (a+x^4\right )^4-6 b d \left (a+x^4\right )^5+d \left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (4 (6 a-b) (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^4\right )^{7/2}}{a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) \left (a+x^4\right )+15 b^4 d \left (a+x^4\right )^2-20 b^3 d \left (a+x^4\right )^3+15 b^2 d \left (a+x^4\right )^4-6 b d \left (a+x^4\right )^5+d \left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (20 (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \left (\frac {a \left (a-b+x^4\right )^{7/2}}{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 ) d-\left (1-6 (a-b)^5 d\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 ) d x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) d x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^{16}+6 a \left (1-\frac {b}{a}\right ) d x^{20}+d x^{24}}+\frac {x^4 \left (a-b+x^4\right )^{7/2}}{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 ) d-\left (1-6 (a-b)^5 d\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 ) d x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) d x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^{16}+6 a \left (1-\frac {b}{a}\right ) d x^{20}+d x^{24}}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (4 (6 a-b) (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^4\right )^{7/2}}{a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) \left (a+x^4\right )+15 b^4 d \left (a+x^4\right )^2-20 b^3 d \left (a+x^4\right )^3+15 b^2 d \left (a+x^4\right )^4-6 b d \left (a+x^4\right )^5+d \left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (20 (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a-b+x^4\right )^{7/2}}{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 ) d-\left (1-6 (a-b)^5 d\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 ) d x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) d x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^{16}+6 a \left (1-\frac {b}{a}\right ) d x^{20}+d x^{24}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (20 a (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^4\right )^{7/2}}{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 ) d-\left (1-6 (a-b)^5 d\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 ) d x^8+20 a^3 \left (1-\frac {b \left (3 a^2-3 a b+b^2\right )}{a^3}\right ) d x^{12}+15 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^{16}+6 a \left (1-\frac {b}{a}\right ) d x^{20}+d x^{24}} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (4 (6 a-b) (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^4\right )^{7/2}}{a \left (1+\frac {b^6 d}{a}\right )-\left (1+6 b^5 d\right ) \left (a+x^4\right )+15 b^4 d \left (a+x^4\right )^2-20 b^3 d \left (a+x^4\right )^3+15 b^2 d \left (a+x^4\right )^4-6 b d \left (a+x^4\right )^5+d \left (a+x^4\right )^6} \, dx,x,\sqrt [4]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 2.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-6 a+b+5 x) \left (-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right )}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^6 d-\left (1+6 b^5 d\right ) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-6*a + b + 5*x)*(-b^5 + 5*b^4*x - 10*b^3*x^2 + 10*b^2*x^3 - 5*b*x^4 + x^5))/(((-a + x)*(-b + x)^2)
^(3/4)*(a + b^6*d - (1 + 6*b^5*d)*x + 15*b^4*d*x^2 - 20*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*x^6)),x]

[Out]

Integrate[((-6*a + b + 5*x)*(-b^5 + 5*b^4*x - 10*b^3*x^2 + 10*b^2*x^3 - 5*b*x^4 + x^5))/(((-a + x)*(-b + x)^2)
^(3/4)*(a + b^6*d - (1 + 6*b^5*d)*x + 15*b^4*d*x^2 - 20*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*x^6)), x]

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IntegrateAlgebraic [A]  time = 0.47, size = 113, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{a-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{a-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-6*a + b + 5*x)*(-b^5 + 5*b^4*x - 10*b^3*x^2 + 10*b^2*x^3 - 5*b*x^4 + x^5))/(((-a + x)*(-
b + x)^2)^(3/4)*(a + b^6*d - (1 + 6*b^5*d)*x + 15*b^4*d*x^2 - 20*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*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)^(3/4))/(a - x)])/d^(3/4) + (2*ArcTanh[
(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/(a - x)])/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((-6*a+b+5*x)*(-b^5+5*b^4*x-10*b^3*x^2+10*b^2*x^3-5*b*x^4+x^5)/((-a+x)*(-b+x)^2)^(3/4)/(a+b^6*d-(6*b^
5*d+1)*x+15*b^4*d*x^2-20*b^3*d*x^3+15*b^2*d*x^4-6*b*d*x^5+d*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 (b^{5} - 5 \, b^{4} x + 10 \, b^{3} x^{2} - 10 \, b^{2} x^{3} + 5 \, b x^{4} - x^{5}\right )} {\left (6 \, a - b - 5 \, x\right )}}{{\left (b^{6} d + 15 \, b^{4} d x^{2} - 20 \, b^{3} d x^{3} + 15 \, b^{2} d x^{4} - 6 \, b d x^{5} + d x^{6} - {\left (6 \, b^{5} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*a+b+5*x)*(-b^5+5*b^4*x-10*b^3*x^2+10*b^2*x^3-5*b*x^4+x^5)/((-a+x)*(-b+x)^2)^(3/4)/(a+b^6*d-(6*b^
5*d+1)*x+15*b^4*d*x^2-20*b^3*d*x^3+15*b^2*d*x^4-6*b*d*x^5+d*x^6),x, algorithm="giac")

[Out]

integrate((b^5 - 5*b^4*x + 10*b^3*x^2 - 10*b^2*x^3 + 5*b*x^4 - x^5)*(6*a - b - 5*x)/((b^6*d + 15*b^4*d*x^2 - 2
0*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*x^6 - (6*b^5*d + 1)*x + a)*(-(a - x)*(b - x)^2)^(3/4)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-6 a +b +5 x \right ) \left (-b^{5}+5 b^{4} x -10 b^{3} x^{2}+10 b^{2} x^{3}-5 b \,x^{4}+x^{5}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (a +b^{6} d -\left (6 b^{5} d +1\right ) x +15 b^{4} d \,x^{2}-20 b^{3} d \,x^{3}+15 b^{2} d \,x^{4}-6 b d \,x^{5}+d \,x^{6}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*a+b+5*x)*(-b^5+5*b^4*x-10*b^3*x^2+10*b^2*x^3-5*b*x^4+x^5)/((-a+x)*(-b+x)^2)^(3/4)/(a+b^6*d-(6*b^5*d+1)
*x+15*b^4*d*x^2-20*b^3*d*x^3+15*b^2*d*x^4-6*b*d*x^5+d*x^6),x)

[Out]

int((-6*a+b+5*x)*(-b^5+5*b^4*x-10*b^3*x^2+10*b^2*x^3-5*b*x^4+x^5)/((-a+x)*(-b+x)^2)^(3/4)/(a+b^6*d-(6*b^5*d+1)
*x+15*b^4*d*x^2-20*b^3*d*x^3+15*b^2*d*x^4-6*b*d*x^5+d*x^6),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b^{5} - 5 \, b^{4} x + 10 \, b^{3} x^{2} - 10 \, b^{2} x^{3} + 5 \, b x^{4} - x^{5}\right )} {\left (6 \, a - b - 5 \, x\right )}}{{\left (b^{6} d + 15 \, b^{4} d x^{2} - 20 \, b^{3} d x^{3} + 15 \, b^{2} d x^{4} - 6 \, b d x^{5} + d x^{6} - {\left (6 \, b^{5} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*a+b+5*x)*(-b^5+5*b^4*x-10*b^3*x^2+10*b^2*x^3-5*b*x^4+x^5)/((-a+x)*(-b+x)^2)^(3/4)/(a+b^6*d-(6*b^
5*d+1)*x+15*b^4*d*x^2-20*b^3*d*x^3+15*b^2*d*x^4-6*b*d*x^5+d*x^6),x, algorithm="maxima")

[Out]

integrate((b^5 - 5*b^4*x + 10*b^3*x^2 - 10*b^2*x^3 + 5*b*x^4 - x^5)*(6*a - b - 5*x)/((b^6*d + 15*b^4*d*x^2 - 2
0*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*x^6 - (6*b^5*d + 1)*x + a)*(-(a - x)*(b - x)^2)^(3/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (b-6\,a+5\,x\right )\,\left (b^5-5\,b^4\,x+10\,b^3\,x^2-10\,b^2\,x^3+5\,b\,x^4-x^5\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (a+b^6\,d+d\,x^6-x\,\left (6\,d\,b^5+1\right )+15\,b^2\,d\,x^4-20\,b^3\,d\,x^3+15\,b^4\,d\,x^2-6\,b\,d\,x^5\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b - 6*a + 5*x)*(5*b*x^4 - 5*b^4*x + b^5 - x^5 - 10*b^2*x^3 + 10*b^3*x^2))/((-(a - x)*(b - x)^2)^(3/4)*(
a + b^6*d + d*x^6 - x*(6*b^5*d + 1) + 15*b^2*d*x^4 - 20*b^3*d*x^3 + 15*b^4*d*x^2 - 6*b*d*x^5)),x)

[Out]

int(-((b - 6*a + 5*x)*(5*b*x^4 - 5*b^4*x + b^5 - x^5 - 10*b^2*x^3 + 10*b^3*x^2))/((-(a - x)*(b - x)^2)^(3/4)*(
a + b^6*d + d*x^6 - x*(6*b^5*d + 1) + 15*b^2*d*x^4 - 20*b^3*d*x^3 + 15*b^4*d*x^2 - 6*b*d*x^5)), x)

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sympy [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((-6*a+b+5*x)*(-b**5+5*b**4*x-10*b**3*x**2+10*b**2*x**3-5*b*x**4+x**5)/((-a+x)*(-b+x)**2)**(3/4)/(a+b
**6*d-(6*b**5*d+1)*x+15*b**4*d*x**2-20*b**3*d*x**3+15*b**2*d*x**4-6*b*d*x**5+d*x**6),x)

[Out]

Timed out

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