3.20.97 \(\int \frac {x^2 (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4)}{(x (-a+x) (-b+x)^3)^{3/4} (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4)} \, dx\)

Optimal. Leaf size=141 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}\right )}{d^{3/4}} \]

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Rubi [F]  time = 29.43, 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 {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x*(-a + x)*(-b + x)^3)^(3/4)*(a*b^3 - b^2*(
3*a + b)*x + 3*b*(a + b)*x^2 - (a + 3*b + d)*x^3 + x^4)),x]

[Out]

(4*(b - x)^2*x*(1 - x/a)^(3/4)*(1 - x/b)^(1/4)*AppellF1[1/4, 3/4, 1/4, 5/4, x/a, x/b])/((a - x)*(b - x)^3*x)^(
3/4) - (4*a*b^3*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(9/4)*Defer[Subst][Defer[Int][1/((-a + x^4)^(3/4)*(-b + x^4)^(
1/4)*(a*b^3 - 3*a*b^2*(1 + b/(3*a))*x^4 + 3*a*b*(1 + b/a)*x^8 - a*(1 + (3*b + d)/a)*x^12 + x^16)), x], x, x^(1
/4)])/((a - x)*(b - x)^3*x)^(3/4) + (4*b^2*(3*a + b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(9/4)*Defer[Subst][Defer[
Int][x^4/((-a + x^4)^(3/4)*(-b + x^4)^(1/4)*(a*b^3 - 3*a*b^2*(1 + b/(3*a))*x^4 + 3*a*b*(1 + b/a)*x^8 - a*(1 +
(3*b + d)/a)*x^12 + x^16)), x], x, x^(1/4)])/((a - x)*(b - x)^3*x)^(3/4) - (12*b*(2*a + b)*x^(3/4)*(-a + x)^(3
/4)*(-b + x)^(9/4)*Defer[Subst][Defer[Int][x^8/((-a + x^4)^(3/4)*(-b + x^4)^(1/4)*(a*b^3 - 3*a*b^2*(1 + b/(3*a
))*x^4 + 3*a*b*(1 + b/a)*x^8 - a*(1 + (3*b + d)/a)*x^12 + x^16)), x], x, x^(1/4)])/((a - x)*(b - x)^3*x)^(3/4)
+ (4*(a + 5*b + d)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(9/4)*Defer[Subst][Defer[Int][x^12/((-a + x^4)^(3/4)*(-b +
x^4)^(1/4)*(a*b^3 - 3*a*b^2*(1 + b/(3*a))*x^4 + 3*a*b*(1 + b/a)*x^8 - a*(1 + (3*b + d)/a)*x^12 + x^16)), x],
x, x^(1/4)])/((a - x)*(b - x)^3*x)^(3/4)

Rubi steps

\begin {align*} \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \int \frac {x^{5/4} \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{(-a+x)^{3/4} (-b+x)^{9/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \int \frac {x^{5/4} \left (3 a b^2+\left (-3 a b-2 b^2\right ) x+b x^2+x^3\right )}{(-a+x)^{3/4} (-b+x)^{5/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \int \frac {x^{5/4} \left (-3 a b+2 b x+x^2\right )}{(-a+x)^{3/4} \sqrt [4]{-b+x} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (-3 a b+2 b x^4+x^8\right )}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-b^2 (3 a+b) x^4+3 b (a+b) x^8-(a+3 b+d) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4}}-\frac {a b^3-b^2 (3 a+b) x^4+3 b (2 a+b) x^8-(a+5 b+d) x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-b^2 (3 a+b) x^4+3 b (a+b) x^8-(a+3 b+d) x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {a b^3-b^2 (3 a+b) x^4+3 b (2 a+b) x^8-(a+5 b+d) x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-b^2 (3 a+b) x^4+3 b (a+b) x^8-(a+3 b+d) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=-\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \left (\frac {a b^3}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}+\frac {(-3 a-b) b^2 x^4}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}+\frac {3 b (2 a+b) x^8}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}+\frac {(-a-5 b-d) x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 x^{3/4} (-b+x)^{9/4} \left (1-\frac {x}{a}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+x^4} \left (1-\frac {x^4}{a}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=-\frac {\left (4 a b^3 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (12 b (2 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 b^2 (3 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (4 (-a-5 b-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 x^{3/4} (-b+x)^2 \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^4}{a}\right )^{3/4} \sqrt [4]{1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ &=\frac {4 (b-x)^2 x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1-\frac {x}{b}} F_1\left (\frac {1}{4};\frac {3}{4},\frac {1}{4};\frac {5}{4};\frac {x}{a},\frac {x}{b}\right )}{\left ((a-x) (b-x)^3 x\right )^{3/4}}-\frac {\left (4 a b^3 x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (12 b (2 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}+\frac {\left (4 b^2 (3 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}-\frac {\left (4 (-a-5 b-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\left (-a+x^4\right )^{3/4} \sqrt [4]{-b+x^4} \left (a b^3-3 a b^2 \left (1+\frac {b}{3 a}\right ) x^4+3 a b \left (1+\frac {b}{a}\right ) x^8-a \left (1+\frac {3 b+d}{a}\right ) x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 4.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x*(-a + x)*(-b + x)^3)^(3/4)*(a*b^3 -
b^2*(3*a + b)*x + 3*b*(a + b)*x^2 - (a + 3*b + d)*x^3 + x^4)),x]

[Out]

Integrate[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x*(-a + x)*(-b + x)^3)^(3/4)*(a*b^3 -
b^2*(3*a + b)*x + 3*b*(a + b)*x^2 - (a + 3*b + d)*x^3 + x^4)), x]

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IntegrateAlgebraic [A]  time = 3.93, size = 141, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x*(-a + x)*(-b + x)^3)^(3/4)
*(a*b^3 - b^2*(3*a + b)*x + 3*b*(a + b)*x^2 - (a + 3*b + d)*x^3 + x^4)),x]

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(3*a*b^3 - 2*(3*a + b)*b^2*x + 3*(a + b)*b*x^2 - x^4)*x^2/(((a - x)*(b - x)^3*x)^(3/4)*(a*b^3 - (3*
a + b)*b^2*x + 3*(a + b)*b*x^2 - (a + 3*b + d)*x^3 + x^4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-3*a*b^3+2*b^2*(3*a+b)*x-3*b*(a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^3)^(3/4)/(a*b^3-b^2*(3*a+b)*x+3*b*(a+b)*
x^2-(a+3*b+d)*x^3+x^4),x)

[Out]

int(x^2*(-3*a*b^3+2*b^2*(3*a+b)*x-3*b*(a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^3)^(3/4)/(a*b^3-b^2*(3*a+b)*x+3*b*(a+b)*
x^2-(a+3*b+d)*x^3+x^4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((3*a*b^3 - 2*(3*a + b)*b^2*x + 3*(a + b)*b*x^2 - x^4)*x^2/(((a - x)*(b - x)^3*x)^(3/4)*(a*b^3 - (3*
a + b)*b^2*x + 3*(a + b)*b*x^2 - (a + 3*b + d)*x^3 + x^4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(3*a*b^3 - x^4 - 2*b^2*x*(3*a + b) + 3*b*x^2*(a + b)))/((x*(a - x)*(b - x)^3)^(3/4)*(a*b^3 - x^3*(a
+ 3*b + d) + x^4 - b^2*x*(3*a + b) + 3*b*x^2*(a + b))),x)

[Out]

int(-(x^2*(3*a*b^3 - x^4 - 2*b^2*x*(3*a + b) + 3*b*x^2*(a + b)))/((x*(a - x)*(b - x)^3)^(3/4)*(a*b^3 - x^3*(a
+ 3*b + d) + x^4 - b^2*x*(3*a + b) + 3*b*x^2*(a + b))), 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(x**2*(-3*a*b**3+2*b**2*(3*a+b)*x-3*b*(a+b)*x**2+x**4)/(x*(-a+x)*(-b+x)**3)**(3/4)/(a*b**3-b**2*(3*a+
b)*x+3*b*(a+b)*x**2-(a+3*b+d)*x**3+x**4),x)

[Out]

Timed out

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