3.8.15 \(\int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx\)
Optimal. Leaf size=198 \[ \frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{12 a^2 c^2 x^2}+\frac {\left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )}{24 a^3 c^3 x}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3} \]
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Rubi [A] time = 0.13, antiderivative size = 198, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used =
{103, 151, 12, 93, 208} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )}{24 a^3 c^3 x}+\frac {\left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{12 a^2 c^2 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^4*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
[Out]
-(Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*x^3) + (5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(12*a^2*c^2*x^2) - ((
15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^3*c^3*x) + ((b*c + a*d)*(5*b^2*c^2 -
2*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(7/2))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}-\frac {\int \frac {\frac {5}{2} (b c+a d)+2 b d x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}+\frac {\int \frac {\frac {1}{4} \left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right )+\frac {5}{2} b d (b c+a d) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^2 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}-\frac {\int \frac {3 (b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}-\frac {\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}-\frac {\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}+\frac {(b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 162, normalized size = 0.82 \begin {gather*} \frac {\left (5 a^3 d^3+3 a^2 b c d^2+3 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )+2 a b c x (7 d x-5 c)+15 b^2 c^2 x^2\right )}{24 a^3 c^3 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^4*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
[Out]
-1/24*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^2*c^2*x^2 + 2*a*b*c*x*(-5*c + 7*d*x) + a^2*(8*c^2 - 10*c*d*x + 15*d^2
*x^2)))/(a^3*c^3*x^3) + ((5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x
])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(7/2))
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IntegrateAlgebraic [A] time = 0.45, size = 359, normalized size = 1.81 \begin {gather*} \frac {\left (5 a^3 d^3+3 a^2 b c d^2+3 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{8 a^{7/2} c^{7/2}}-\frac {\sqrt {c+d x} \left (\frac {15 a^5 d^3 (c+d x)^2}{(a+b x)^2}-\frac {40 a^4 c d^3 (c+d x)}{a+b x}+\frac {9 a^4 b c d^2 (c+d x)^2}{(a+b x)^2}+\frac {9 a^3 b^2 c^2 d (c+d x)^2}{(a+b x)^2}-\frac {24 a^3 b c^2 d^2 (c+d x)}{a+b x}+33 a^3 c^2 d^3-\frac {33 a^2 b^3 c^3 (c+d x)^2}{(a+b x)^2}+\frac {24 a^2 b^2 c^3 d (c+d x)}{a+b x}-9 a^2 b c^3 d^2+\frac {40 a b^3 c^4 (c+d x)}{a+b x}-9 a b^2 c^4 d-15 b^3 c^5\right )}{24 a^3 c^3 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x^4*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
[Out]
-1/24*(Sqrt[c + d*x]*(-15*b^3*c^5 - 9*a*b^2*c^4*d - 9*a^2*b*c^3*d^2 + 33*a^3*c^2*d^3 + (40*a*b^3*c^4*(c + d*x)
)/(a + b*x) + (24*a^2*b^2*c^3*d*(c + d*x))/(a + b*x) - (24*a^3*b*c^2*d^2*(c + d*x))/(a + b*x) - (40*a^4*c*d^3*
(c + d*x))/(a + b*x) - (33*a^2*b^3*c^3*(c + d*x)^2)/(a + b*x)^2 + (9*a^3*b^2*c^2*d*(c + d*x)^2)/(a + b*x)^2 +
(9*a^4*b*c*d^2*(c + d*x)^2)/(a + b*x)^2 + (15*a^5*d^3*(c + d*x)^2)/(a + b*x)^2))/(a^3*c^3*Sqrt[a + b*x]*(-c +
(a*(c + d*x))/(a + b*x))^3) + ((5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*ArcTanh[(Sqrt[a]*Sqrt[c
+ d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(8*a^(7/2)*c^(7/2))
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fricas [A] time = 2.31, size = 436, normalized size = 2.20 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c^{4} x^{3}}, -\frac {3 \, {\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c^{4} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")
[Out]
[1/96*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(a*c)*x^3*log((8*a^2*c^2 + (b^2*c^2 + 6*a
*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d
)*x)/x^2) - 4*(8*a^3*c^3 + (15*a*b^2*c^3 + 14*a^2*b*c^2*d + 15*a^3*c*d^2)*x^2 - 10*(a^2*b*c^3 + a^3*c^2*d)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^3), -1/48*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*s
qrt(-a*c)*x^3*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2
+ (a*b*c^2 + a^2*c*d)*x)) + 2*(8*a^3*c^3 + (15*a*b^2*c^3 + 14*a^2*b*c^2*d + 15*a^3*c*d^2)*x^2 - 10*(a^2*b*c^3
+ a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^3)]
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giac [B] time = 30.54, size = 1932, normalized size = 9.76
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")
[Out]
1/24*sqrt(b*d)*b^8*d^3*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*arctan(-1/2*(b^2*c + a*b*d -
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^3*b^
7*c^3*d^3) - 2*(15*b^13*c^8 - 76*a*b^12*c^7*d + 156*a^2*b^11*c^6*d^2 - 180*a^3*b^10*c^5*d^3 + 170*a^4*b^9*c^4*
d^4 - 180*a^5*b^8*c^3*d^5 + 156*a^6*b^7*c^2*d^6 - 76*a^7*b^6*c*d^7 + 15*a^8*b^5*d^8 - 75*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^11*c^7 + 135*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^2*a*b^10*c^6*d + 45*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^9*c^
5*d^2 - 105*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^8*c^4*d^3 - 105*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^7*c^3*d^4 + 45*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^6*c^2*d^5 + 135*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^2*a^6*b^5*c*d^6 - 75*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^4*d^7 +
150*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^9*c^6 + 60*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^8*c^5*d + 42*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a^2*b^7*c^4*d^2 - 504*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^6*c^
3*d^3 + 42*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^5*c^2*d^4 + 60*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b^4*c*d^5 + 150*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^4*a^6*b^3*d^6 - 150*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^6*b^7*c^5 - 230*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^6*c^4*d - 324*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^5*c^3*d^2 - 324*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^4*c^2*d^3 - 230*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^6*a^4*b^3*c*d^4 - 150*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^2*d^
5 + 75*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^5*c^4 + 120*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^4*c^3*d + 90*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^8*a^2*b^3*c^2*d^2 + 120*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^3*b^
2*c*d^3 + 75*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^4*b*d^4 - 15*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^3*c^3 - 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^10*a*b^2*c^2*d - 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b*c
*d^2 - 15*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*d^3)/((b^4*c^2 - 2*a*b^3*c*d
+ a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^4)^3*a^3*b^6*c^3*d^3))/abs(b)
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maple [B] time = 0.03, size = 408, normalized size = 2.06 \begin {gather*} \frac {\left (15 a^{3} d^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+9 a^{2} b c \,d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+9 a \,b^{2} c^{2} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+15 b^{3} c^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} d^{2} x^{2}-28 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a b c d \,x^{2}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{2} c^{2} x^{2}+20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} c d x +20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a b \,c^{2} x -16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} c^{2}\right ) \sqrt {d x +c}\, \sqrt {b x +a}}{48 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^4/(b*x+a)^(1/2)/(d*x+c)^(1/2),x)
[Out]
1/48/a^3/c^3*(15*a^3*d^3*x^3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+9*a^2*b*c*d^2*x^3
*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+9*a*b^2*c^2*d*x^3*ln((a*d*x+b*c*x+2*a*c+2*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+15*b^3*c^3*x^3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
))/x)-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*d^2*x^2-28*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b*c*d*x^2-30
*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^2*c^2*x^2+20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*c*d*x+20*((b*x+a)*
(d*x+c))^(1/2)*(a*c)^(1/2)*a*b*c^2*x-16*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*c^2)*(d*x+c)^(1/2)*(b*x+a)^(1/
2)/(a*c)^(1/2)/x^3/((b*x+a)*(d*x+c))^(1/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?
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mupad [B] time = 90.68, size = 1518, normalized size = 7.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^4*(a + b*x)^(1/2)*(c + d*x)^(1/2)),x)
[Out]
(((a + b*x)^(1/2) - a^(1/2))*((7*b*d^2)/(64*a^2*c^2) - (d*(a*d + b*c)^2)/(4*a^3*c^3) + (d*(3*a^2*d^2 + 3*b^2*c
^2 + 8*a*b*c*d))/(32*a^3*c^3)))/((c + d*x)^(1/2) - c^(1/2)) - (b^6/(192*a^2*c^2*d^3) + (((a + b*x)^(1/2) - a^(
1/2))^2*((5*b^6*c^2)/64 + (5*a^2*b^4*d^2)/64))/(a^3*c^3*d^3*((c + d*x)^(1/2) - c^(1/2))^2) + (((a + b*x)^(1/2)
- a^(1/2))^4*((13*b^6*c^4)/64 + (13*a^4*b^2*d^4)/64 + (55*a^3*b^3*c*d^3)/32 + (57*a^2*b^4*c^2*d^2)/32 + (55*a
*b^5*c^3*d)/32))/(a^4*c^4*d^3*((c + d*x)^(1/2) - c^(1/2))^4) - (((a + b*x)^(1/2) - a^(1/2))^5*((39*a^4*b^2*c*d
^4)/32 - (17*a^5*b*d^5)/64 - (17*b^6*c^5)/64 + (25*a^2*b^4*c^3*d^2)/8 + (25*a^3*b^3*c^2*d^3)/8 + (39*a*b^5*c^4
*d)/32))/(a^(9/2)*c^(9/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^5) - (((a + b*x)^(1/2) - a^(1/2))^7*((97*a^2*b^3*c^3
*d^2)/64 - (27*b^5*c^5)/64 - (27*a^5*d^5)/64 + (97*a^3*b^2*c^2*d^3)/64 + (15*a*b^4*c^4*d)/64 + (15*a^4*b*c*d^4
)/64))/(a^(9/2)*c^(9/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^7) + (((b^6*c)/64 + (a*b^5*d)/64)*((a + b*x)^(1/2) - a
^(1/2)))/(a^(5/2)*c^(5/2)*d^3*((c + d*x)^(1/2) - c^(1/2))) - (((a + b*x)^(1/2) - a^(1/2))^3*((73*b^6*c^3)/192
+ (73*a^3*b^3*d^3)/192 + (9*a^2*b^4*c*d^2)/16 + (9*a*b^5*c^2*d)/16))/(a^(7/2)*c^(7/2)*d^3*((c + d*x)^(1/2) - c
^(1/2))^3) - (((a + b*x)^(1/2) - a^(1/2))^6*((37*a^6*d^6)/192 + (37*b^6*c^6)/192 - (71*a^2*b^4*c^4*d^2)/32 - (
49*a^3*b^3*c^3*d^3)/16 - (71*a^4*b^2*c^2*d^4)/32 + (11*a*b^5*c^5*d)/32 + (11*a^5*b*c*d^5)/32))/(a^5*c^5*d^3*((
c + d*x)^(1/2) - c^(1/2))^6) + (((a + b*x)^(1/2) - a^(1/2))^8*((15*a^2*b^2*c^2*d^2)/64 - (15*b^4*c^4)/64 - (15
*a^4*d^4)/64 + (11*a*b^3*c^3*d)/32 + (11*a^3*b*c*d^3)/32))/(a^4*c^4*d*((c + d*x)^(1/2) - c^(1/2))^8))/(((a + b
*x)^(1/2) - a^(1/2))^9/((c + d*x)^(1/2) - c^(1/2))^9 + (b^3*((a + b*x)^(1/2) - a^(1/2))^3)/(d^3*((c + d*x)^(1/
2) - c^(1/2))^3) + (((a + b*x)^(1/2) - a^(1/2))^7*(3*a^2*d^2 + 3*b^2*c^2 + 9*a*b*c*d))/(a*c*d^2*((c + d*x)^(1/
2) - c^(1/2))^7) - (((a + b*x)^(1/2) - a^(1/2))^6*(a^3*d^3 + b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2))/(a^(3/2
)*c^(3/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^6) - ((3*a*d + 3*b*c)*((a + b*x)^(1/2) - a^(1/2))^8)/(a^(1/2)*c^(1/2
)*d*((c + d*x)^(1/2) - c^(1/2))^8) - ((3*b^3*c + 3*a*b^2*d)*((a + b*x)^(1/2) - a^(1/2))^4)/(a^(1/2)*c^(1/2)*d^
3*((c + d*x)^(1/2) - c^(1/2))^4) + (((a + b*x)^(1/2) - a^(1/2))^5*(3*b^3*c^2 + 3*a^2*b*d^2 + 9*a*b^2*c*d))/(a*
c*d^3*((c + d*x)^(1/2) - c^(1/2))^5)) - (log(((c^(1/2)*(a + b*x)^(1/2) - a^(1/2)*(c + d*x)^(1/2))*(b*c^(1/2) -
(a^(1/2)*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))))/((c + d*x)^(1/2) - c^(1/2)))*(5*a^(1/2)
*b^3*c^(7/2) + 5*a^(7/2)*c^(1/2)*d^3 + 3*a^(3/2)*b^2*c^(5/2)*d + 3*a^(5/2)*b*c^(3/2)*d^2))/(16*a^4*c^4) + (log
(((a + b*x)^(1/2) - a^(1/2))/((c + d*x)^(1/2) - c^(1/2)))*(5*a^(1/2)*b^3*c^(7/2) + 5*a^(7/2)*c^(1/2)*d^3 + 3*a
^(3/2)*b^2*c^(5/2)*d + 3*a^(5/2)*b*c^(3/2)*d^2))/(16*a^4*c^4) - (d^3*((a + b*x)^(1/2) - a^(1/2))^3)/(192*a^2*c
^2*((c + d*x)^(1/2) - c^(1/2))^3) - (d^2*(a*d + b*c)*((a + b*x)^(1/2) - a^(1/2))^2)/(32*a^(5/2)*c^(5/2)*((c +
d*x)^(1/2) - c^(1/2))^2)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt {a + b x} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**4/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
[Out]
Integral(1/(x**4*sqrt(a + b*x)*sqrt(c + d*x)), x)
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