3.652 \(\int \frac {1}{x^3 \sqrt {a+b \sqrt {c+d x}}} \, dx\)

Optimal. Leaf size=261 \[ -\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 x^2 \left (a^2-b^2 c\right )}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c x \left (a^2-b^2 c\right )^2}+\frac {b d^2 \left (2 a-5 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 c^{3/2} \left (a-b \sqrt {c}\right )^{5/2}}-\frac {b d^2 \left (2 a+5 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 c^{3/2} \left (a+b \sqrt {c}\right )^{5/2}} \]

[Out]

1/16*b*d^2*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a-b*c^(1/2))^(1/2))*(2*a-5*b*c^(1/2))/c^(3/2)/(a-b*c^(1/2))^(5/2
)-1/16*b*d^2*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a+b*c^(1/2))^(1/2))*(2*a+5*b*c^(1/2))/c^(3/2)/(a+b*c^(1/2))^(5
/2)-1/2*(a-b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))^(1/2)/(-b^2*c+a^2)/x^2-1/8*b*d*(6*a*b*c-(5*b^2*c+a^2)*(d*x+c)^
(1/2))*(a+b*(d*x+c)^(1/2))^(1/2)/c/(-b^2*c+a^2)^2/x

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Rubi [A]  time = 0.48, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {371, 1398, 823, 827, 1166, 207} \[ -\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 x^2 \left (a^2-b^2 c\right )}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c x \left (a^2-b^2 c\right )^2}+\frac {b d^2 \left (2 a-5 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 c^{3/2} \left (a-b \sqrt {c}\right )^{5/2}}-\frac {b d^2 \left (2 a+5 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 c^{3/2} \left (a+b \sqrt {c}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/(2*(a^2 - b^2*c)*x^2) - (b*d*Sqrt[a + b*Sqrt[c + d*x]]*(6*a
*b*c - (a^2 + 5*b^2*c)*Sqrt[c + d*x]))/(8*c*(a^2 - b^2*c)^2*x) + (b*(2*a - 5*b*Sqrt[c])*d^2*ArcTanh[Sqrt[a + b
*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(16*(a - b*Sqrt[c])^(5/2)*c^(3/2)) - (b*(2*a + 5*b*Sqrt[c])*d^2*ArcTanh[
Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/(16*(a + b*Sqrt[c])^(5/2)*c^(3/2))

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \sqrt {a+b \sqrt {c+d x}}} \, dx &=d^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {x}} (-c+x)^3} \, dx,x,c+d x\right )\\ &=\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x} \left (-c+x^2\right )^3} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}+\frac {d^2 \operatorname {Subst}\left (\int \frac {-\frac {1}{2} a b c+\frac {5}{2} b^2 c x}{\sqrt {a+b x} \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} a b c \left (a^2-4 b^2 c\right )+\frac {1}{4} b^2 c \left (a^2+5 b^2 c\right ) x}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{4 c^2 \left (a^2-b^2 c\right )^2}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} a b^2 c \left (a^2-4 b^2 c\right )-\frac {1}{4} a b^2 c \left (a^2+5 b^2 c\right )+\frac {1}{4} b^2 c \left (a^2+5 b^2 c\right ) x^2}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 c^2 \left (a^2-b^2 c\right )^2}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}-\frac {\left (b \left (2 a-5 b \sqrt {c}\right ) d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{16 \left (a-b \sqrt {c}\right )^2 c^{3/2}}+\frac {\left (b \left (2 a+5 b \sqrt {c}\right ) d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{16 \left (a+b \sqrt {c}\right )^2 c^{3/2}}\\ &=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \sqrt {a+b \sqrt {c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt {c+d x}\right )}{8 c \left (a^2-b^2 c\right )^2 x}+\frac {b \left (2 a-5 b \sqrt {c}\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 \left (a-b \sqrt {c}\right )^{5/2} c^{3/2}}-\frac {b \left (2 a+5 b \sqrt {c}\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 \left (a+b \sqrt {c}\right )^{5/2} c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.75, size = 281, normalized size = 1.08 \[ -\frac {\frac {8 \left (a^2-b^2 c\right ) \left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{x^2}-\frac {2 b d \sqrt {a+b \sqrt {c+d x}} \left (a^2 \sqrt {c+d x}-6 a b c+5 b^2 c \sqrt {c+d x}\right )}{c x}+\frac {b d^2 \left (\left (a-b \sqrt {c}\right )^{5/2} \left (2 a+5 b \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )-\left (2 a-5 b \sqrt {c}\right ) \left (a+b \sqrt {c}\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )\right )}{c^{3/2} \sqrt {a-b \sqrt {c}} \sqrt {a+b \sqrt {c}}}}{16 \left (a^2-b^2 c\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*((8*(a^2 - b^2*c)*(a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/x^2 - (2*b*d*Sqrt[a + b*Sqrt[c + d*x]
]*(-6*a*b*c + a^2*Sqrt[c + d*x] + 5*b^2*c*Sqrt[c + d*x]))/(c*x) + (b*d^2*(-((2*a - 5*b*Sqrt[c])*(a + b*Sqrt[c]
)^(5/2)*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]]) + (a - b*Sqrt[c])^(5/2)*(2*a + 5*b*Sqrt[c])*Ar
cTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]]))/(Sqrt[a - b*Sqrt[c]]*Sqrt[a + b*Sqrt[c]]*c^(3/2)))/(a^2
- b^2*c)^2

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fricas [B]  time = 1.28, size = 4390, normalized size = 16.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*((b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)*x^2*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2
)*d^4 + (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18
*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18
*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a
^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*
a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))*log((625*b^12*c^3 + 3750*a^2*b^10*c^2 - 1491*a^4*b^8*c + 140*a^6*b^6)
*sqrt(sqrt(d*x + c)*b + a)*d^6 + ((325*a*b^12*c^5 + 1977*a^3*b^10*c^4 - 609*a^5*b^8*c^3 + 35*a^7*b^6*c^2)*d^4
- (5*b^14*c^10 - 16*a^2*b^12*c^9 + 3*a^4*b^10*c^8 + 50*a^6*b^8*c^7 - 85*a^8*b^6*c^6 + 60*a^10*b^4*c^5 - 19*a^1
2*b^2*c^4 + 2*a^14*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*
a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^
10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))*sqrt(-((10
5*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 + (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10
*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a
^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*
b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*
c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))) - (b^4*c^3 -
2*a^2*b^2*c^2 + a^4*c)*x^2*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 + (b^10*c^8
- 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^
16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^1
6*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a
^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^
8*b^2*c^4 - a^10*c^3))*log((625*b^12*c^3 + 3750*a^2*b^10*c^2 - 1491*a^4*b^8*c + 140*a^6*b^6)*sqrt(sqrt(d*x + c
)*b + a)*d^6 - ((325*a*b^12*c^5 + 1977*a^3*b^10*c^4 - 609*a^5*b^8*c^3 + 35*a^7*b^6*c^2)*d^4 - (5*b^14*c^10 - 1
6*a^2*b^12*c^9 + 3*a^4*b^10*c^8 + 50*a^6*b^8*c^7 - 85*a^8*b^6*c^6 + 60*a^10*b^4*c^5 - 19*a^12*b^2*c^4 + 2*a^14
*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^2
0*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*
a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))*sqrt(-((105*a*b^8*c^3 + 70*a
^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 + (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a
^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a
^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^1
0*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 -
5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))) + (b^4*c^3 - 2*a^2*b^2*c^2 + a^
4*c)*x^2*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 - (b^10*c^8 - 5*a^2*b^8*c^7 +
10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4
*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b
^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a
^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c
^3))*log((625*b^12*c^3 + 3750*a^2*b^10*c^2 - 1491*a^4*b^8*c + 140*a^6*b^6)*sqrt(sqrt(d*x + c)*b + a)*d^6 + ((3
25*a*b^12*c^5 + 1977*a^3*b^10*c^4 - 609*a^5*b^8*c^3 + 35*a^7*b^6*c^2)*d^4 + (5*b^14*c^10 - 16*a^2*b^12*c^9 + 3
*a^4*b^10*c^8 + 50*a^6*b^8*c^7 - 85*a^8*b^6*c^6 + 60*a^10*b^4*c^5 - 19*a^12*b^2*c^4 + 2*a^14*c^3)*sqrt((625*b^
18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^
18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120
*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^
5*b^4*c + 4*a^7*b^2)*d^4 - (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*
c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20
*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a
^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 1
0*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))) - (b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)*x^2*sqrt(-((1
05*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 - (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 1
0*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*
a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8
*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20
*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))*log((625*b^12
*c^3 + 3750*a^2*b^10*c^2 - 1491*a^4*b^8*c + 140*a^6*b^6)*sqrt(sqrt(d*x + c)*b + a)*d^6 - ((325*a*b^12*c^5 + 19
77*a^3*b^10*c^4 - 609*a^5*b^8*c^3 + 35*a^7*b^6*c^2)*d^4 + (5*b^14*c^10 - 16*a^2*b^12*c^9 + 3*a^4*b^10*c^8 + 50
*a^6*b^8*c^7 - 85*a^8*b^6*c^6 + 60*a^10*b^4*c^5 - 19*a^12*b^2*c^4 + 2*a^14*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*
b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b
^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45
*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^
2)*d^4 - (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^1
8*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^1
8*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*
a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10
*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))) + 4*(6*a*b^2*c*d*x - 4*a*b^2*c^2 + 4*a^3*c + (4*b^3*c^2 - 4*a^2*b*c
- (5*b^3*c + a^2*b)*d*x)*sqrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a))/((b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)*x^2)

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giac [B]  time = 1.27, size = 1303, normalized size = 4.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(((b^5*c^3 - 2*a^2*b^3*c^2 + a^4*b*c)^2*(5*b^6*c + a^2*b^4)*d^3 - (13*a*b^10*c^(7/2) - 27*a^3*b^8*c^(5/2)
+ 15*a^5*b^6*c^(3/2) - a^7*b^4*sqrt(c))*d^3*abs(b^5*c^3 - 2*a^2*b^3*c^2 + a^4*b*c) + 2*(4*a^2*b^14*c^6 - 17*a
^4*b^12*c^5 + 28*a^6*b^10*c^4 - 22*a^8*b^8*c^3 + 8*a^10*b^6*c^2 - a^12*b^4*c)*d^3)*arctan(sqrt(sqrt(d*x + c)*b
+ a)/sqrt(-(a*b^4*c^3 - 2*a^3*b^2*c^2 + a^5*c + sqrt((a*b^4*c^3 - 2*a^3*b^2*c^2 + a^5*c)^2 + (b^6*c^4 - 3*a^2
*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)))/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)))/((b^
9*c^6 - a*b^8*c^(11/2) - 4*a^2*b^7*c^5 + 4*a^3*b^6*c^(9/2) + 6*a^4*b^5*c^4 - 6*a^5*b^4*c^(7/2) - 4*a^6*b^3*c^3
+ 4*a^7*b^2*c^(5/2) + a^8*b*c^2 - a^9*c^(3/2))*sqrt(-b*sqrt(c) - a)*abs(b^5*c^3 - 2*a^2*b^3*c^2 + a^4*b*c)) +
((b^5*c^3 - 2*a^2*b^3*c^2 + a^4*b*c)^2*(5*b^6*c + a^2*b^4)*d^3 + (13*a*b^10*c^(7/2) - 27*a^3*b^8*c^(5/2) + 15
*a^5*b^6*c^(3/2) - a^7*b^4*sqrt(c))*d^3*abs(b^5*c^3 - 2*a^2*b^3*c^2 + a^4*b*c) + 2*(4*a^2*b^14*c^6 - 17*a^4*b^
12*c^5 + 28*a^6*b^10*c^4 - 22*a^8*b^8*c^3 + 8*a^10*b^6*c^2 - a^12*b^4*c)*d^3)*arctan(sqrt(sqrt(d*x + c)*b + a)
/sqrt(-(a*b^4*c^3 - 2*a^3*b^2*c^2 + a^5*c - sqrt((a*b^4*c^3 - 2*a^3*b^2*c^2 + a^5*c)^2 + (b^6*c^4 - 3*a^2*b^4*
c^3 + 3*a^4*b^2*c^2 - a^6*c)*(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)))/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)))/((b^9*c^6
+ a*b^8*c^(11/2) - 4*a^2*b^7*c^5 - 4*a^3*b^6*c^(9/2) + 6*a^4*b^5*c^4 + 6*a^5*b^4*c^(7/2) - 4*a^6*b^3*c^3 - 4*
a^7*b^2*c^(5/2) + a^8*b*c^2 + a^9*c^(3/2))*sqrt(b*sqrt(c) - a)*abs(b^5*c^3 - 2*a^2*b^3*c^2 + a^4*b*c)) - 2*(9*
(sqrt(d*x + c)*b + a)^(3/2)*b^8*c^2*d^3 - 19*sqrt(sqrt(d*x + c)*b + a)*a*b^8*c^2*d^3 - 5*(sqrt(d*x + c)*b + a)
^(7/2)*b^6*c*d^3 + 21*(sqrt(d*x + c)*b + a)^(5/2)*a*b^6*c*d^3 - 30*(sqrt(d*x + c)*b + a)^(3/2)*a^2*b^6*c*d^3 +
18*sqrt(sqrt(d*x + c)*b + a)*a^3*b^6*c*d^3 - (sqrt(d*x + c)*b + a)^(7/2)*a^2*b^4*d^3 + 3*(sqrt(d*x + c)*b + a
)^(5/2)*a^3*b^4*d^3 - 3*(sqrt(d*x + c)*b + a)^(3/2)*a^4*b^4*d^3 + sqrt(sqrt(d*x + c)*b + a)*a^5*b^4*d^3)/((b^4
*c^3 - 2*a^2*b^2*c^2 + a^4*c)*(b^2*c - (sqrt(d*x + c)*b + a)^2 + 2*(sqrt(d*x + c)*b + a)*a - a^2)^2))/(b^2*d)

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maple [B]  time = 0.10, size = 840, normalized size = 3.22 \[ \frac {a \,b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{8 \sqrt {b^{2} c}\, \left (b^{2} c +a^{2}+2 \sqrt {b^{2} c}\, a \right ) \sqrt {-a -\sqrt {b^{2} c}}\, c}+\frac {a \,b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{8 \sqrt {b^{2} c}\, \left (-b^{2} c -a^{2}+2 \sqrt {b^{2} c}\, a \right ) \sqrt {-a +\sqrt {b^{2} c}}\, c}+\frac {\left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} a \,b^{2} d^{2}}{8 \sqrt {b^{2} c}\, \left (\sqrt {d x +c}\, b -\sqrt {b^{2} c}\right )^{2} \left (b^{2} c +a^{2}+2 \sqrt {b^{2} c}\, a \right ) c}-\frac {\sqrt {a +\sqrt {d x +c}\, b}\, a \,b^{2} d^{2}}{8 \sqrt {b^{2} c}\, \left (\sqrt {d x +c}\, b -\sqrt {b^{2} c}\right )^{2} \left (a +\sqrt {b^{2} c}\right ) c}-\frac {\left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} a \,b^{2} d^{2}}{8 \sqrt {b^{2} c}\, \left (\sqrt {d x +c}\, b +\sqrt {b^{2} c}\right )^{2} \left (b^{2} c +a^{2}-2 \sqrt {b^{2} c}\, a \right ) c}+\frac {\sqrt {a +\sqrt {d x +c}\, b}\, a \,b^{2} d^{2}}{8 \sqrt {b^{2} c}\, \left (\sqrt {d x +c}\, b +\sqrt {b^{2} c}\right )^{2} \left (a -\sqrt {b^{2} c}\right ) c}+\frac {5 b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{16 \left (b^{2} c +a^{2}+2 \sqrt {b^{2} c}\, a \right ) \sqrt {-a -\sqrt {b^{2} c}}\, c}-\frac {5 b^{2} d^{2} \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{16 \left (-b^{2} c -a^{2}+2 \sqrt {b^{2} c}\, a \right ) \sqrt {-a +\sqrt {b^{2} c}}\, c}+\frac {5 \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} b^{2} d^{2}}{16 \left (\sqrt {d x +c}\, b -\sqrt {b^{2} c}\right )^{2} \left (b^{2} c +a^{2}+2 \sqrt {b^{2} c}\, a \right ) c}-\frac {7 \sqrt {a +\sqrt {d x +c}\, b}\, b^{2} d^{2}}{16 \left (\sqrt {d x +c}\, b -\sqrt {b^{2} c}\right )^{2} \left (a +\sqrt {b^{2} c}\right ) c}+\frac {5 \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} b^{2} d^{2}}{16 \left (\sqrt {d x +c}\, b +\sqrt {b^{2} c}\right )^{2} \left (b^{2} c +a^{2}-2 \sqrt {b^{2} c}\, a \right ) c}-\frac {7 \sqrt {a +\sqrt {d x +c}\, b}\, b^{2} d^{2}}{16 \left (\sqrt {d x +c}\, b +\sqrt {b^{2} c}\right )^{2} \left (a -\sqrt {b^{2} c}\right ) c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*b^2*d^2/c/((d*x+c)^(1/2)*b-(b^2*c)^(1/2))^2/(b^2*c+2*a*(b^2*c)^(1/2)+a^2)*(a+(d*x+c)^(1/2)*b)^(3/2)+1/8*b
^2*d^2/c/(b^2*c)^(1/2)/((d*x+c)^(1/2)*b-(b^2*c)^(1/2))^2/(b^2*c+2*a*(b^2*c)^(1/2)+a^2)*(a+(d*x+c)^(1/2)*b)^(3/
2)*a-7/16*b^2*d^2/c/((d*x+c)^(1/2)*b-(b^2*c)^(1/2))^2/((b^2*c)^(1/2)+a)*(a+(d*x+c)^(1/2)*b)^(1/2)-1/8*b^2*d^2/
c/(b^2*c)^(1/2)/((d*x+c)^(1/2)*b-(b^2*c)^(1/2))^2/((b^2*c)^(1/2)+a)*(a+(d*x+c)^(1/2)*b)^(1/2)*a+5/16*b^2*d^2/c
/(b^2*c+2*a*(b^2*c)^(1/2)+a^2)/(-a-(b^2*c)^(1/2))^(1/2)*arctan((a+(d*x+c)^(1/2)*b)^(1/2)/(-a-(b^2*c)^(1/2))^(1
/2))+1/8*b^2*d^2/c/(b^2*c)^(1/2)/(b^2*c+2*a*(b^2*c)^(1/2)+a^2)/(-a-(b^2*c)^(1/2))^(1/2)*arctan((a+(d*x+c)^(1/2
)*b)^(1/2)/(-a-(b^2*c)^(1/2))^(1/2))*a+5/16*b^2*d^2/c/((d*x+c)^(1/2)*b+(b^2*c)^(1/2))^2/(b^2*c-2*a*(b^2*c)^(1/
2)+a^2)*(a+(d*x+c)^(1/2)*b)^(3/2)-1/8*b^2*d^2/c/(b^2*c)^(1/2)/((d*x+c)^(1/2)*b+(b^2*c)^(1/2))^2/(b^2*c-2*a*(b^
2*c)^(1/2)+a^2)*(a+(d*x+c)^(1/2)*b)^(3/2)*a+1/8*b^2*d^2/c/(b^2*c)^(1/2)/((d*x+c)^(1/2)*b+(b^2*c)^(1/2))^2/(-(b
^2*c)^(1/2)+a)*(a+(d*x+c)^(1/2)*b)^(1/2)*a-7/16*b^2*d^2/c/((d*x+c)^(1/2)*b+(b^2*c)^(1/2))^2/(-(b^2*c)^(1/2)+a)
*(a+(d*x+c)^(1/2)*b)^(1/2)-5/16*b^2*d^2/c/(-b^2*c+2*a*(b^2*c)^(1/2)-a^2)/(-a+(b^2*c)^(1/2))^(1/2)*arctan((a+(d
*x+c)^(1/2)*b)^(1/2)/(-a+(b^2*c)^(1/2))^(1/2))+1/8*b^2*d^2/c/(b^2*c)^(1/2)/(-b^2*c+2*a*(b^2*c)^(1/2)-a^2)/(-a+
(b^2*c)^(1/2))^(1/2)*arctan((a+(d*x+c)^(1/2)*b)^(1/2)/(-a+(b^2*c)^(1/2))^(1/2))*a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sqrt {d x + c} b + a} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(sqrt(d*x + c)*b + a)*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^3\,\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt {a + b \sqrt {c + d x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**3*sqrt(a + b*sqrt(c + d*x))), x)

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