3.16.51 \(\int \frac {x^2 (3 a b^2-2 b (2 a+b) x+(a+2 b) x^2)}{(x (-a+x) (-b+x)^2)^{3/4} (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3)} \, dx\)

Optimal. Leaf size=107 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}\right )}{d^{3/4}} \]

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Rubi [F]  time = 19.18, 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^2-2 b (2 a+b) x+(a+2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*(a + 2*b)*((b*(a - x))/(a*(b - x)))^(3/4)*(b - x)^2*x*Hypergeometric2F1[1/4, 3/4, 5/4, -(((a - b)*x)/(a*(b
- x)))])/(b*(1 - d)*(-((a - x)*(b - x)^2*x))^(3/4)) + (4*a*b^2*(a + 2*b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)
*Defer[Subst][Defer[Int][1/((-a + x^4)^(3/4)*Sqrt[-b + x^4]*(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4
)^2)), x], x, x^(1/4)])/((1 - d)*(-((a - x)*(b - x)^2*x))^(3/4)) - (4*b*(2*a + b)*(a + 2*b)*x^(3/4)*(-a + x)^(
3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][x^4/((-a + x^4)^(3/4)*Sqrt[-b + x^4]*(-(b^2*x^4) + 2*b*x^8 + (-1 +
d)*x^12 + a*(b - x^4)^2)), x], x, x^(1/4)])/((1 - d)*(-((a - x)*(b - x)^2*x))^(3/4)) + (4*(a^2 + 4*b^2 + a*(b
+ 3*b*d))*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][x^8/((-a + x^4)^(3/4)*Sqrt[-b + x^4]*
(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4)^2)), x], x, x^(1/4)])/((1 - d)*(-((a - x)*(b - x)^2*x))^(3
/4))

Rubi steps

\begin {align*} \int \frac {x^2 \left (3 a b^2-2 b (2 a+b) x+(a+2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {x^{5/4} \left (3 a b^2-2 b (2 a+b) x+(a+2 b) x^2\right )}{(-a+x)^{3/4} (-b+x)^{3/2} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {x^{5/4} (-3 a b+(a+2 b) x)}{(-a+x)^{3/4} \sqrt {-b+x} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (-3 a b+(a+2 b) x^4\right )}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-b (2 a+b) x^4+(a+2 b) x^8+(-1+d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \left (-\frac {a+2 b}{(1-d) \left (-a+x^4\right )^{3/4} \sqrt {-b+x^4}}-\frac {a b^2 (a+2 b)-b (2 a+b) (a+2 b) x^4+\left (a^2+4 b^2+a (b+3 b d)\right ) x^8}{(-1+d) \left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-b (2 a+b) x^4+(a+2 b) x^8+(-1+d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=-\frac {\left (4 (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4}} \, dx,x,\sqrt [4]{x}\right )}{(1-d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {a b^2 (a+2 b)-b (2 a+b) (a+2 b) x^4+\left (a^2+4 b^2+a (b+3 b d)\right ) x^8}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-b (2 a+b) x^4+(a+2 b) x^8+(-1+d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {4 (a+2 b) \left (\frac {b (a-x)}{a (b-x)}\right )^{3/4} (b-x)^2 x \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {(a-b) x}{a (b-x)}\right )}{b (1-d) \left (-\left ((a-x) (b-x)^2 x\right )\right )^{3/4}}-\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \left (\frac {a b^2 (a+2 b)}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}\right )}+\frac {(-a-2 b) b (2 a+b) x^4}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}\right )}+\frac {\left (a^2+4 b^2+a (b+3 b d)\right ) x^8}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {4 (a+2 b) \left (\frac {b (a-x)}{a (b-x)}\right )^{3/4} (b-x)^2 x \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {(a-b) x}{a (b-x)}\right )}{b (1-d) \left (-\left ((a-x) (b-x)^2 x\right )\right )^{3/4}}-\frac {\left (4 a b^2 (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (4 b (2 a+b) (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (4 \left (a^2+4 b^2+a (b+3 b d)\right ) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {4 (a+2 b) \left (\frac {b (a-x)}{a (b-x)}\right )^{3/4} (b-x)^2 x \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {(a-b) x}{a (b-x)}\right )}{b (1-d) \left (-\left ((a-x) (b-x)^2 x\right )\right )^{3/4}}-\frac {\left (4 a b^2 (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (4 b (2 a+b) (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (4 \left (a^2+4 b^2+a (b+3 b d)\right ) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \sqrt {-b+x^4} \left (-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d) \left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 4.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (3 a b^2-2 b (2 a+b) x+(a+2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.64, size = 107, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((3*a*b^2 - 2*(2*a + b)*b*x + (a + 2*b)*x^2)*x^2/((-(a - x)*(b - x)^2*x)^(3/4)*((d - 1)*x^3 + a*b^2 -
(2*a + b)*b*x + (a + 2*b)*x^2)), 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^2+x^2\,\left (a+2\,b\right )-2\,b\,x\,\left (2\,a+b\right )\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (a\,b^2+x^2\,\left (a+2\,b\right )+x^3\,\left (d-1\right )-b\,x\,\left (2\,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^2 + x^2*(a + 2*b) - 2*b*x*(2*a + b)))/((-x*(a - x)*(b - x)^2)^(3/4)*(a*b^2 + x^2*(a + 2*b) + x
^3*(d - 1) - b*x*(2*a + b))),x)

[Out]

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

[Out]

Timed out

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