3.342 \(\int \frac {\sqrt {-c+d x} \sqrt {c+d x} (a+b x^2)}{x^5} \, dx\)

Optimal. Leaf size=121 \[ -\frac {\sqrt {d x-c} \sqrt {c+d x} \left (a d^2+4 b c^2\right )}{8 c^2 x^2}+\frac {d^2 \left (a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )}{8 c^3}+\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4} \]

[Out]

1/4*a*(d*x-c)^(3/2)*(d*x+c)^(3/2)/c^2/x^4+1/8*d^2*(a*d^2+4*b*c^2)*arctan((d*x-c)^(1/2)*(d*x+c)^(1/2)/c)/c^3-1/
8*(a*d^2+4*b*c^2)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/c^2/x^2

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Rubi [A]  time = 0.10, antiderivative size = 164, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {454, 94, 92, 205} \[ -\frac {\sqrt {d x-c} (c+d x)^{3/2} \left (a d^2+4 b c^2\right )}{8 c^3 x^2}+\frac {d \sqrt {d x-c} \sqrt {c+d x} \left (a d^2+4 b c^2\right )}{8 c^3 x}+\frac {d^2 \left (a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )}{8 c^3}+\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x^5,x]

[Out]

(d*(4*b*c^2 + a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(8*c^3*x) - ((4*b*c^2 + a*d^2)*Sqrt[-c + d*x]*(c + d*x)^(3/
2))/(8*c^3*x^2) + (a*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(4*c^2*x^4) + (d^2*(4*b*c^2 + a*d^2)*ArcTan[(Sqrt[-c +
d*x]*Sqrt[c + d*x])/c])/(8*c^3)

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 454

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*e*
(m + 1)), x] + Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*(m + 1)), Int[(e*x)^(m + n)*(a1
+ b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && Eq
Q[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1
])) &&  !ILtQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 137, normalized size = 1.13 \[ \frac {\sqrt {d x-c} \sqrt {c+d x} \left (\left (c^2-d^2 x^2\right ) \left (2 a c^2-a d^2 x^2+4 b c^2 x^2\right )-d^2 x^4 \sqrt {1-\frac {d^2 x^2}{c^2}} \left (a d^2+4 b c^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {d^2 x^2}{c^2}}\right )\right )}{8 c^2 d^2 x^6-8 c^4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-c + d*x]*Sqrt[c + d*x]*((c^2 - d^2*x^2)*(2*a*c^2 + 4*b*c^2*x^2 - a*d^2*x^2) - d^2*(4*b*c^2 + a*d^2)*x^4
*Sqrt[1 - (d^2*x^2)/c^2]*ArcTanh[Sqrt[1 - (d^2*x^2)/c^2]]))/(-8*c^4*x^4 + 8*c^2*d^2*x^6)

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fricas [A]  time = 1.09, size = 100, normalized size = 0.83 \[ \frac {2 \, {\left (4 \, b c^{2} d^{2} + a d^{4}\right )} x^{4} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right ) - {\left (2 \, a c^{3} + {\left (4 \, b c^{3} - a c d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{8 \, c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(4*b*c^2*d^2 + a*d^4)*x^4*arctan(-(d*x - sqrt(d*x + c)*sqrt(d*x - c))/c) - (2*a*c^3 + (4*b*c^3 - a*c*d^
2)*x^2)*sqrt(d*x + c)*sqrt(d*x - c))/(c^3*x^4)

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giac [B]  time = 0.45, size = 324, normalized size = 2.68 \[ -\frac {\frac {{\left (4 \, b c^{2} d^{3} + a d^{5}\right )} \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} - \frac {2 \, {\left (4 \, b c^{2} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} - a d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 16 \, b c^{4} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} + 28 \, a c^{2} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} - 64 \, b c^{6} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 112 \, a c^{4} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 256 \, b c^{8} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 64 \, a c^{6} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{2}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/x^5,x, algorithm="giac")

[Out]

-1/4*((4*b*c^2*d^3 + a*d^5)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c)/c^3 - 2*(4*b*c^2*d^3*(sqrt(d*x + c
) - sqrt(d*x - c))^14 - a*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^14 + 16*b*c^4*d^3*(sqrt(d*x + c) - sqrt(d*x - c)
)^10 + 28*a*c^2*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^10 - 64*b*c^6*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^6 - 112*
a*c^4*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^6 - 256*b*c^8*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^2 + 64*a*c^6*d^5*(
sqrt(d*x + c) - sqrt(d*x - c))^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^4*c^2))/d

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maple [B]  time = 0.07, size = 226, normalized size = 1.87 \[ -\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (a \,d^{4} x^{4} \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right )+4 b \,c^{2} d^{2} x^{4} \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right )-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{2} x^{2}+4 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{2} x^{2}+2 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, a \,c^{2}\right )}{8 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(d*x-c)^(1/2)*(d*x+c)^(1/2)/c^2*(ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*x^4*a*d^4+4*ln(-2*(c^2-(
-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*x^4*b*c^2*d^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2)*x^2*a*d^2+4*(-c^2)^(1/2)*(d
^2*x^2-c^2)^(1/2)*x^2*b*c^2+2*(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2)*a*c^2)/(d^2*x^2-c^2)^(1/2)/x^4/(-c^2)^(1/2)

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maxima [A]  time = 1.35, size = 162, normalized size = 1.34 \[ -\frac {b d^{2} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{2 \, c} - \frac {a d^{4} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{8 \, c^{3}} - \frac {\sqrt {d^{2} x^{2} - c^{2}} b d^{2}}{2 \, c^{2}} - \frac {\sqrt {d^{2} x^{2} - c^{2}} a d^{4}}{8 \, c^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b}{2 \, c^{2} x^{2}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a d^{2}}{8 \, c^{4} x^{2}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a}{4 \, c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/x^5,x, algorithm="maxima")

[Out]

-1/2*b*d^2*arcsin(c/(d*abs(x)))/c - 1/8*a*d^4*arcsin(c/(d*abs(x)))/c^3 - 1/2*sqrt(d^2*x^2 - c^2)*b*d^2/c^2 - 1
/8*sqrt(d^2*x^2 - c^2)*a*d^4/c^4 + 1/2*(d^2*x^2 - c^2)^(3/2)*b/(c^2*x^2) + 1/8*(d^2*x^2 - c^2)^(3/2)*a*d^2/(c^
4*x^2) + 1/4*(d^2*x^2 - c^2)^(3/2)*a/(c^2*x^4)

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mupad [B]  time = 15.56, size = 1004, normalized size = 8.30 \[ \frac {\frac {a\,\sqrt {-c}\,d^4}{1024\,c^{7/2}}+\frac {a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{128\,c^{7/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {11\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{512\,c^{7/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}+\frac {7\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{256\,c^{7/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}-\frac {239\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{1024\,c^{7/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}+\frac {a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}{256\,c^{7/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{10}}}{\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}+\frac {4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}+\frac {4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{10}}+\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{12}}}-\frac {\frac {b\,\sqrt {-c}\,d^2}{32\,c^{3/2}}+\frac {b\,\sqrt {-c}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{16\,c^{3/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}-\frac {15\,b\,\sqrt {-c}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{32\,c^{3/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}}{\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}+\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}}+\frac {a\,\sqrt {-c}\,d^4\,\ln \left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{8\,c^{7/2}}+\frac {b\,\sqrt {-c}\,d^2\,\ln \left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{2\,c^{3/2}}-\frac {a\,\sqrt {-c}\,d^4\,\ln \left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )}{8\,c^{7/2}}-\frac {b\,\sqrt {-c}\,d^2\,\ln \left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )}{2\,c^{3/2}}+\frac {a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{256\,c^{7/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{1024\,c^{7/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {b\,\sqrt {-c}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{32\,c^{3/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*(-c)^(1/2)*d^4)/(1024*c^(7/2)) + (a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^2)/(128*c^(7/2)*((-c)^(1/2)
- (d*x - c)^(1/2))^2) + (11*a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^4)/(512*c^(7/2)*((-c)^(1/2) - (d*x -
c)^(1/2))^4) + (7*a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^6)/(256*c^(7/2)*((-c)^(1/2) - (d*x - c)^(1/2))
^6) - (239*a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^8)/(1024*c^(7/2)*((-c)^(1/2) - (d*x - c)^(1/2))^8) + (
a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^10)/(256*c^(7/2)*((-c)^(1/2) - (d*x - c)^(1/2))^10))/(((c + d*x)^
(1/2) - c^(1/2))^4/((-c)^(1/2) - (d*x - c)^(1/2))^4 + (4*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c
)^(1/2))^6 + (6*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 + (4*((c + d*x)^(1/2) - c^(1/2
))^10)/((-c)^(1/2) - (d*x - c)^(1/2))^10 + ((c + d*x)^(1/2) - c^(1/2))^12/((-c)^(1/2) - (d*x - c)^(1/2))^12) -
((b*(-c)^(1/2)*d^2)/(32*c^(3/2)) + (b*(-c)^(1/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^2)/(16*c^(3/2)*((-c)^(1/2) -
(d*x - c)^(1/2))^2) - (15*b*(-c)^(1/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^4)/(32*c^(3/2)*((-c)^(1/2) - (d*x - c)
^(1/2))^4))/(((c + d*x)^(1/2) - c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (2*((c + d*x)^(1/2) - c^(1/2))^4
)/((-c)^(1/2) - (d*x - c)^(1/2))^4 + ((c + d*x)^(1/2) - c^(1/2))^6/((-c)^(1/2) - (d*x - c)^(1/2))^6) + (a*(-c)
^(1/2)*d^4*log(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))/(8*c^(7/2)) + (b*(-c)^(1/2)*d^2*lo
g(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))/(2*c^(3/2)) - (a*(-c)^(1/2)*d^4*log(((c + d*x)^
(1/2) - c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + 1))/(8*c^(7/2)) - (b*(-c)^(1/2)*d^2*log(((c + d*x)^(1/2)
- c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + 1))/(2*c^(3/2)) + (a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2
))^2)/(256*c^(7/2)*((-c)^(1/2) - (d*x - c)^(1/2))^2) + (a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^4)/(1024*
c^(7/2)*((-c)^(1/2) - (d*x - c)^(1/2))^4) - (b*(-c)^(1/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^2)/(32*c^(3/2)*((-c)
^(1/2) - (d*x - c)^(1/2))^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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