3.27.34 \(\int \frac {1}{(-b x+a^2 x^2)^{3/2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\) [2634]

Optimal. Leaf size=232 \[ \frac {2 \sqrt {-b x+a^2 x^2} \left (210 b^3-200 a^2 b^2 x-456 a^4 b x^2+1601 a^6 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{1155 b^5 x^4 \left (b-a^2 x\right )}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (-\frac {4 \left (245 a b^2+461 a^3 b x+2533 a^5 x^2\right )}{1155 b^5 x^3}-\frac {6 a^{11/2} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{b^{11/2} x}\right ) \]

[Out]

2/1155*(a^2*x^2-b*x)^(1/2)*(1601*a^6*x^3-456*a^4*b*x^2-200*a^2*b^2*x+210*b^3)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1
/2)/b^5/x^4/(-a^2*x+b)+(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)*(-4/1155*(2533*a^5*x^2+461*a^3*b*x+245*a*b^2)/b^5/x
^3-6*a^(11/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)*arctan(a^(1/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)/b^(1/2))/b^(11/
2)/x)

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Rubi [F]
time = 2.68, 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 {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a^2 x}\right ) \int \frac {1}{x^{3/2} \left (-b+a^2 x\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx}{\sqrt {-b x+a^2 x^2}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b+a^2 x}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (-b+a^2 x^2\right )^{3/2} \left (a x^4+x^2 \sqrt {-b x^2+a^2 x^4}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 4.62, size = 236, normalized size = 1.02 \begin {gather*} -\frac {2 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (\sqrt {b} \left (210 b^3+10 a b^2 \left (-20 a x+49 \sqrt {x \left (-b+a^2 x\right )}\right )+2 a^3 b x \left (-228 a x+461 \sqrt {x \left (-b+a^2 x\right )}\right )+a^5 x^2 \left (1601 a x+5066 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+3465 a^{11/2} x^2 \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{1155 b^{11/2} x^3 \sqrt {x \left (-b+a^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(Sqrt[b]*(210*b^3 + 10*a*b^2*(-20*a*x + 49*Sqrt[x*(-b + a^2*x)]) + 2*
a^3*b*x*(-228*a*x + 461*Sqrt[x*(-b + a^2*x)]) + a^5*x^2*(1601*a*x + 5066*Sqrt[x*(-b + a^2*x)])) + 3465*a^(11/2
)*x^2*Sqrt[x*(-b + a^2*x)]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]]*ArcTan[(Sqrt[a]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*
x)]])/Sqrt[b]]))/(1155*b^(11/2)*x^3*Sqrt[x*(-b + a^2*x)])

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{2} x^{2}-b x \right )^{\frac {3}{2}} \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.51, size = 438, normalized size = 1.89 \begin {gather*} \left [\frac {3465 \, {\left (a^{7} x^{5} - a^{5} b x^{4}\right )} \sqrt {a} \log \left (\frac {a^{2} x^{2} + 2 \, \sqrt {a^{2} x^{2} - b x} a x - b x + 2 \, \sqrt {a^{2} x^{2} - b x} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {a}}{a^{2} x^{2} - b x}\right ) - 2 \, {\left (5066 \, a^{7} x^{4} - 4144 \, a^{5} b x^{3} - 432 \, a^{3} b^{2} x^{2} - 490 \, a b^{3} x + {\left (1601 \, a^{6} x^{3} - 456 \, a^{4} b x^{2} - 200 \, a^{2} b^{2} x + 210 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{1155 \, {\left (a^{2} b^{5} x^{5} - b^{6} x^{4}\right )}}, -\frac {2 \, {\left (3465 \, {\left (a^{7} x^{5} - a^{5} b x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{a x}\right ) + {\left (5066 \, a^{7} x^{4} - 4144 \, a^{5} b x^{3} - 432 \, a^{3} b^{2} x^{2} - 490 \, a b^{3} x + {\left (1601 \, a^{6} x^{3} - 456 \, a^{4} b x^{2} - 200 \, a^{2} b^{2} x + 210 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}\right )}}{1155 \, {\left (a^{2} b^{5} x^{5} - b^{6} x^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1155*(3465*(a^7*x^5 - a^5*b*x^4)*sqrt(a)*log((a^2*x^2 + 2*sqrt(a^2*x^2 - b*x)*a*x - b*x + 2*sqrt(a^2*x^2 -
b*x)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(a))/(a^2*x^2 - b*x)) - 2*(5066*a^7*x^4 - 4144*a^5*b*x^3 - 432*a^
3*b^2*x^2 - 490*a*b^3*x + (1601*a^6*x^3 - 456*a^4*b*x^2 - 200*a^2*b^2*x + 210*b^3)*sqrt(a^2*x^2 - b*x))*sqrt(a
*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^2*b^5*x^5 - b^6*x^4), -2/1155*(3465*(a^7*x^5 - a^5*b*x^4)*sqrt(-a)*arctan(sq
rt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(-a)/(a*x)) + (5066*a^7*x^4 - 4144*a^5*b*x^3 - 432*a^3*b^2*x^2 - 490*a*b
^3*x + (1601*a^6*x^3 - 456*a^4*b*x^2 - 200*a^2*b^2*x + 210*b^3)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2
- b*x)*x))/(a^2*b^5*x^5 - b^6*x^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x*(a*x + sqrt(a**2*x**2 - b*x)))**(3/2)*(x*(a**2*x - b))**(3/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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