3.29.35 \(\int \frac {1}{(-b x+a^2 x^2)^{5/2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\)

Optimal. Leaf size=288 \[ \frac {\sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (121339 a^{10} x^5-148243 a^8 b x^4+12416 a^6 b^2 x^3+5248 a^4 b^3 x^2+2688 a^2 b^4 x-4368 b^5\right )}{16380 b^7 x^5 \left (b-a^2 x\right )^2}+\sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (\frac {109 a^{15/2} \sqrt {\sqrt {a^2 x^2-b x}-a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\sqrt {a^2 x^2-b x}-a x}}{\sqrt {b}}\right )}{4 b^{15/2} x}-\frac {283847 a^9 x^4-229768 a^7 b x^3-24840 a^5 b^2 x^2-9352 a^3 b^3 x-4872 a b^4}{8190 b^7 x^4 \left (b-a^2 x\right )}\right ) \]

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Rubi [F]  time = 4.14, 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 )^{5/2} \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)^(5/2)*(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^4*(-b + a^2*x^2)^(5/2)*(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 )^{5/2} \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^{5/2} \left (-b+a^2 x\right )^{5/2} \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 ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (-b+a^2 x^2\right )^{5/2} \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 [F]  time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \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]

Integrate[1/((-(b*x) + a^2*x^2)^(5/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]

[Out]

Integrate[1/((-(b*x) + a^2*x^2)^(5/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)), x]

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IntegrateAlgebraic [A]  time = 8.16, size = 288, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-b x+a^2 x^2} \left (-4368 b^5+2688 a^2 b^4 x+5248 a^4 b^3 x^2+12416 a^6 b^2 x^3-148243 a^8 b x^4+121339 a^{10} x^5\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{16380 b^7 x^5 \left (b-a^2 x\right )^2}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (-\frac {-4872 a b^4-9352 a^3 b^3 x-24840 a^5 b^2 x^2-229768 a^7 b x^3+283847 a^9 x^4}{8190 b^7 x^4 \left (b-a^2 x\right )}+\frac {109 a^{15/2} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{4 b^{15/2} x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((-(b*x) + a^2*x^2)^(5/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]

[Out]

(Sqrt[-(b*x) + a^2*x^2]*(-4368*b^5 + 2688*a^2*b^4*x + 5248*a^4*b^3*x^2 + 12416*a^6*b^2*x^3 - 148243*a^8*b*x^4
+ 121339*a^10*x^5)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(16380*b^7*x^5*(b - a^2*x)^2) + Sqrt[x*(a*x + Sqrt[
-(b*x) + a^2*x^2])]*(-1/8190*(-4872*a*b^4 - 9352*a^3*b^3*x - 24840*a^5*b^2*x^2 - 229768*a^7*b*x^3 + 283847*a^9
*x^4)/(b^7*x^4*(b - a^2*x)) + (109*a^(15/2)*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqrt[a]*Sqrt[-(a*x)
+ Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(4*b^(15/2)*x))

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fricas [A]  time = 0.53, size = 566, normalized size = 1.97 \begin {gather*} \left [\frac {446355 \, {\left (a^{11} x^{7} - 2 \, a^{9} b x^{6} + a^{7} b^{2} x^{5}\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 (567694 \, a^{11} x^{6} - 1027230 \, a^{9} b x^{5} + 409856 \, a^{7} b^{2} x^{4} + 30976 \, a^{5} b^{3} x^{3} + 8960 \, a^{3} b^{4} x^{2} + 9744 \, a b^{5} x + {\left (121339 \, a^{10} x^{5} - 148243 \, a^{8} b x^{4} + 12416 \, a^{6} b^{2} x^{3} + 5248 \, a^{4} b^{3} x^{2} + 2688 \, a^{2} b^{4} x - 4368 \, b^{5}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{32760 \, {\left (a^{4} b^{7} x^{7} - 2 \, a^{2} b^{8} x^{6} + b^{9} x^{5}\right )}}, \frac {446355 \, {\left (a^{11} x^{7} - 2 \, a^{9} b x^{6} + a^{7} b^{2} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{a x}\right ) + {\left (567694 \, a^{11} x^{6} - 1027230 \, a^{9} b x^{5} + 409856 \, a^{7} b^{2} x^{4} + 30976 \, a^{5} b^{3} x^{3} + 8960 \, a^{3} b^{4} x^{2} + 9744 \, a b^{5} x + {\left (121339 \, a^{10} x^{5} - 148243 \, a^{8} b x^{4} + 12416 \, a^{6} b^{2} x^{3} + 5248 \, a^{4} b^{3} x^{2} + 2688 \, a^{2} b^{4} x - 4368 \, b^{5}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{16380 \, {\left (a^{4} b^{7} x^{7} - 2 \, a^{2} b^{8} x^{6} + b^{9} x^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b*x)^(5/2)/(a*x^2+x*(a^2*x^2-b*x)^(1/2))^(3/2),x, algorithm="fricas")

[Out]

[1/32760*(446355*(a^11*x^7 - 2*a^9*b*x^6 + a^7*b^2*x^5)*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*(567694*a^11*x^6 -
1027230*a^9*b*x^5 + 409856*a^7*b^2*x^4 + 30976*a^5*b^3*x^3 + 8960*a^3*b^4*x^2 + 9744*a*b^5*x + (121339*a^10*x^
5 - 148243*a^8*b*x^4 + 12416*a^6*b^2*x^3 + 5248*a^4*b^3*x^2 + 2688*a^2*b^4*x - 4368*b^5)*sqrt(a^2*x^2 - b*x))*
sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^4*b^7*x^7 - 2*a^2*b^8*x^6 + b^9*x^5), 1/16380*(446355*(a^11*x^7 - 2*a^
9*b*x^6 + a^7*b^2*x^5)*sqrt(-a)*arctan(sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(-a)/(a*x)) + (567694*a^11*x^6
- 1027230*a^9*b*x^5 + 409856*a^7*b^2*x^4 + 30976*a^5*b^3*x^3 + 8960*a^3*b^4*x^2 + 9744*a*b^5*x + (121339*a^10*
x^5 - 148243*a^8*b*x^4 + 12416*a^6*b^2*x^3 + 5248*a^4*b^3*x^2 + 2688*a^2*b^4*x - 4368*b^5)*sqrt(a^2*x^2 - b*x)
)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^4*b^7*x^7 - 2*a^2*b^8*x^6 + b^9*x^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b*x)^(5/2)/(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)^(5/2)*(a*x^2 + sqrt(a^2*x^2 - b*x)*x)^(3/2)), x)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{2} x^{2}-b x \right )^{\frac {5}{2}} \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2*x^2-b*x)^(5/2)/(a*x^2+x*(a^2*x^2-b*x)^(1/2))^(3/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b*x)^(5/2)/(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)^(5/2)*(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 )}^{5/2}\,{\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)^(5/2)*(a*x^2 + x*(a^2*x^2 - b*x)^(1/2))^(3/2)),x)

[Out]

int(1/((a^2*x^2 - b*x)^(5/2)*(a*x^2 + x*(a^2*x^2 - b*x)^(1/2))^(3/2)), x)

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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 {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2*x**2-b*x)**(5/2)/(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))**(5/2)), x)

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