3.6.55 \(\int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx\) [555]

Optimal. Leaf size=256 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}+\frac {\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^2 x^2}-\frac {(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^3 x}+\frac {(b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{7/2}} \]

[Out]

1/64*(-a*d+b*c)^2*(5*a^2*d^2+6*a*b*c*d+5*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)
/c^(7/2)-1/4*(b*x+a)^(1/2)*(d*x+c)^(1/2)/x^4-1/24*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c/x^3+1/96*(5*a^2*d^
2-2*a*b*c*d+5*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2/x^2-1/192*(a*d+b*c)*(15*a^2*d^2-22*a*b*c*d+15*b^2*c
^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3/x

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Rubi [A]
time = 0.14, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {99, 156, 12, 95, 214} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )}{96 a^2 c^2 x^2}+\frac {\left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{192 a^3 c^3 x}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{24 a c x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[a + b*x]*Sqrt[c + d*x])/x^4 - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a*c*x^3) + ((5*b^2*c^2
- 2*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a^2*c^2*x^2) - ((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d
+ 15*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^3*c^3*x) + ((b*c - a*d)^2*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^
2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*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 95

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 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 156

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] && ILtQ[m, -1]

Rule 214

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}+\frac {1}{4} \int \frac {\frac {1}{2} (b c+a d)+b d x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}-\frac {\int \frac {\frac {1}{4} \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )+b d (b c+a d) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 a c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}+\frac {\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^2 x^2}+\frac {\int \frac {\frac {1}{8} (b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )+\frac {1}{4} b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^2 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}+\frac {\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^2 x^2}-\frac {(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^3 x}-\frac {\int \frac {3 (b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}+\frac {\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^2 x^2}-\frac {(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^3 x}-\frac {\left ((b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}+\frac {\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^2 x^2}-\frac {(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^3 x}-\frac {\left ((b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}+\frac {\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^2 x^2}-\frac {(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^3 x}+\frac {(b c-a d)^2 \left (5 b^2 c^2+6 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 )}{64 a^{7/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 201, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^3 c^3 x^3-a b^2 c^2 x^2 (10 c+7 d x)+a^2 b c x \left (8 c^2+4 c d x-7 d^2 x^2\right )+a^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )\right )}{192 a^3 c^3 x^4}+\frac {(b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{7/2} c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 - a*b^2*c^2*x^2*(10*c + 7*d*x) + a^2*b*c*x*(8*c^2 + 4*c*d*
x - 7*d^2*x^2) + a^3*(48*c^3 + 8*c^2*d*x - 10*c*d^2*x^2 + 15*d^3*x^3)))/(a^3*c^3*x^4) + ((b*c - a*d)^2*(5*b^2*
c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(64*a^(7/2)*c^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(218)=436\).
time = 0.06, size = 593, normalized size = 2.32

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} d^{3} x^{3}+14 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b c \,d^{2} x^{3}+14 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{2} d \,x^{3}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{3} c^{3} x^{3}+20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c \,d^{2} x^{2}-8 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{2} d \,x^{2}+20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{3} x^{2}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{2} d x -16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{3} x -96 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{3} c^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{4} \sqrt {a c}}\)	\(593\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*
a^4*d^4*x^4-12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b*c*d^3*x^4-6*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b^2*c^2*d^2*x^4-12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+
c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b^3*c^3*d*x^4+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)
*b^4*c^4*x^4-30*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*d^3*x^3+14*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c
*d^2*x^3+14*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^2*c^2*d*x^3-30*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^3*c^3
*x^3+20*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*c*d^2*x^2-8*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c^2*d*x^
2+20*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^2*c^3*x^2-16*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*c^2*d*x-16*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c^3*x-96*((d*x+c)*(b*x+a))^(1/2)*a^3*c^3*(a*c)^(1/2))/((d*x+c)*(b*x+a
))^(1/2)/x^4/(a*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((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5,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 detail

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Fricas [A]
time = 2.55, size = 568, normalized size = 2.22 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 7 \, a^{2} b^{2} c^{3} d - 7 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{4} c^{4} x^{4}}, -\frac {3 \, {\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \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 (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 7 \, a^{2} b^{2} c^{3} d - 7 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{4} c^{4} x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 - 7*a^2*b^2*c^3*d - 7*a^3*b*c^2*d^2 + 15*a^4*c*
d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 2*a^3*b*c^3*d + 5*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/(a^4*c^4*x^4), -1/384*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4
*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 - 7*a^2*b^2*c^3*d - 7*a^3*b*c^2*d^2 + 15*a^4
*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 2*a^3*b*c^3*d + 5*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(a^4*c^4*x^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3834 vs. \(2 (218) = 436\).
time = 4.12, size = 3834, normalized size = 14.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*(5*sqrt(b*d)*b^5*c^4*abs(b) - 4*sqrt(b*d)*a*b^4*c^3*d*abs(b) - 2*sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) - 4
*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) + 5*sqrt(b*d)*a^4*b*d^4*abs(b))*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*c^3) - 2*(15*sqrt(
b*d)*b^19*c^11*abs(b) - 127*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 469*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 965*sqrt(
b*d)*a^3*b^16*c^8*d^3*abs(b) + 1126*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) - 518*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b)
- 518*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) + 1126*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 965*sqrt(b*d)*a^8*b^11*c^3*
d^8*abs(b) + 469*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) - 127*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) + 15*sqrt(b*d)*a^11*
b^8*d^11*abs(b) - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*ab
s(b) + 554*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d*abs(b) - 1
013*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^15*c^8*d^2*abs(b) + 248*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^14*c^7*d^3*abs(b) + 1886*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^13*c^6*d^4*abs(b) - 3140*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^12*c^5*d^5*abs(b) + 1886*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^11*c^4*d^6*abs(b) + 248*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7*abs(b) - 1013*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^9*c^2*d^8*abs(b) + 554*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^9*b^8*c*d^9*abs(b) - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^7*d^10*abs(b) + 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^15*c^9*abs(b) - 877*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^4*a*b^14*c^8*d*abs(b) + 428*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4*a^2*b^13*c^7*d^2*abs(b) + 828*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^4*a^3*b^12*c^6*d^3*abs(b) - 694*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a^4*b^11*c^5*d^4*abs(b) - 694*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^5*b^10*c^4*d^5*abs(b) + 828*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^6*b^9*c^3*d^6*abs(b) + 428*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*
a^7*b^8*c^2*d^7*abs(b) - 877*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^8*b
^7*c*d^8*abs(b) + 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^9*b^6*d^9*
abs(b) - 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^13*c^8*abs(b) + 520
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^12*c^7*d*abs(b) + 212*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^11*c^6*d^2*abs(b) + 824*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^10*c^5*d^3*abs(b) - 2062*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^9*c^4*d^4*abs(b) + 824*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^8*c^3*d^5*abs(b) + 212*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^7*c^2*d^6*abs(b) + 520*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^6*c*d^7*abs(b) - 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^5*d^8*abs(b) + 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^8*b^11*c^7*abs(b) + 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^8*a*b^10*c^6*d*abs(b) - 115*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^8*a^2*b^9*c^5*d^2*abs(b) + 559*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^8*a^3*b^8*c^4*d^3*abs(b) + 559*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*
a^4*b^7*c^3*d^4*abs(b) - 115*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^5*b
^6*c^2*d^5*abs(b) + 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^6*b^5*c*d
^6*abs(b) + 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^7*b^4*d^7*abs(b)
- 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^9*c^6*abs(b) - 158*sqrt(
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^8*c^5*d*abs(b) + 139*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*...

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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