3.6.31 \(\int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx\)

Optimal. Leaf size=345 \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (7 a^2 d^2-2 a b c d+7 b^2 c^2\right )}{240 a^2 c^2 x^3}-\frac {(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{960 a^3 c^3 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^4 x}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{40 a c x^4} \]

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Rubi [A]  time = 0.32, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {97, 151, 12, 93, 208} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{960 a^3 c^3 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (7 a^2 d^2-2 a b c d+7 b^2 c^2\right )}{240 a^2 c^2 x^3}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4-40 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^4 x}-\frac {(a d+b c) \left (7 a^2 d^2+2 a b c d+7 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 )}{128 a^{9/2} c^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{40 a c x^4}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^6,x]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(5*x^5) - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a*c*x^4) + ((7*b^2*c^2
- 2*a*b*c*d + 7*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(240*a^2*c^2*x^3) - ((b*c + a*d)*(35*b^2*c^2 - 46*a*b*c*
d + 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^3*c^3*x^2) + ((105*b^4*c^4 - 40*a*b^3*c^3*d - 34*a^2*b^2*c
^2*d^2 - 40*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^4*c^4*x) - ((b*c - a*d)^2*(b*c + a
*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)
*c^(9/2))

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 97

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 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 {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}+\frac {1}{5} \int \frac {\frac {1}{2} (b c+a d)+b d x}{x^5 \sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}-\frac {\int \frac {\frac {1}{4} \left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right )+\frac {3}{2} b d (b c+a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 a c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}+\frac {\int \frac {\frac {1}{8} (b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right )+\frac {1}{2} b d \left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a^2 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}-\frac {\int \frac {\frac {1}{16} \left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right )+\frac {1}{8} b d (b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}+\frac {\int \frac {15 (b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^4 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}+\frac {\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^4 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}+\frac {\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 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 )}{128 a^4 c^4}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}+\frac {\left (7 b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}-\frac {(b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.61, size = 253, normalized size = 0.73 \begin {gather*} -\frac {\frac {8 x^2 (a+b x)^{3/2} (c+d x)^{3/2} \left (35 a^2 d^2+38 a b c d+35 b^2 c^2\right )}{a^2 c^2}+\frac {15 x^3 (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) \left (x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+a d x+b c x)\right )}{a^{7/2} c^{7/2}}-\frac {336 x (a+b x)^{3/2} (c+d x)^{3/2} (a d+b c)}{a c}+384 (a+b x)^{3/2} (c+d x)^{3/2}}{1920 a c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^6,x]

[Out]

-1/1920*(384*(a + b*x)^(3/2)*(c + d*x)^(3/2) - (336*(b*c + a*d)*x*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(a*c) + (8*
(35*b^2*c^2 + 38*a*b*c*d + 35*a^2*d^2)*x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(a^2*c^2) + (15*(b*c + a*d)*(7*b^2
*c^2 + 2*a*b*c*d + 7*a^2*d^2)*x^3*(-(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c + b*c*x + a*d*x)) + (b
*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(7/2)*c^(7/2)))/(a*c*x^5)

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IntegrateAlgebraic [A]  time = 0.96, size = 583, normalized size = 1.69 \begin {gather*} \frac {\sqrt {c+d x} (a d-b c)^2 \left (\frac {105 a^7 d^3 (c+d x)^4}{(a+b x)^4}-\frac {490 a^6 c d^3 (c+d x)^3}{(a+b x)^3}+\frac {135 a^6 b c d^2 (c+d x)^4}{(a+b x)^4}+\frac {135 a^5 b^2 c^2 d (c+d x)^4}{(a+b x)^4}+\frac {896 a^5 c^2 d^3 (c+d x)^2}{(a+b x)^2}-\frac {630 a^5 b c^2 d^2 (c+d x)^3}{(a+b x)^3}+\frac {105 a^4 b^3 c^3 (c+d x)^4}{(a+b x)^4}-\frac {630 a^4 b^2 c^3 d (c+d x)^3}{(a+b x)^3}-\frac {790 a^4 c^3 d^3 (c+d x)}{a+b x}+\frac {1152 a^4 b c^3 d^2 (c+d x)^2}{(a+b x)^2}+\frac {790 a^3 b^3 c^4 (c+d x)^3}{(a+b x)^3}-\frac {1152 a^3 b^2 c^4 d (c+d x)^2}{(a+b x)^2}+\frac {630 a^3 b c^4 d^2 (c+d x)}{a+b x}-105 a^3 c^4 d^3-\frac {896 a^2 b^3 c^5 (c+d x)^2}{(a+b x)^2}+\frac {630 a^2 b^2 c^5 d (c+d x)}{a+b x}-135 a^2 b c^5 d^2+\frac {490 a b^3 c^6 (c+d x)}{a+b x}-135 a b^2 c^6 d-105 b^3 c^7\right )}{1920 a^4 c^4 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^5}-\frac {(a d-b c)^2 \left (7 a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+7 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{9/2} c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^6,x]

[Out]

((-(b*c) + a*d)^2*Sqrt[c + d*x]*(-105*b^3*c^7 - 135*a*b^2*c^6*d - 135*a^2*b*c^5*d^2 - 105*a^3*c^4*d^3 + (490*a
*b^3*c^6*(c + d*x))/(a + b*x) + (630*a^2*b^2*c^5*d*(c + d*x))/(a + b*x) + (630*a^3*b*c^4*d^2*(c + d*x))/(a + b
*x) - (790*a^4*c^3*d^3*(c + d*x))/(a + b*x) - (896*a^2*b^3*c^5*(c + d*x)^2)/(a + b*x)^2 - (1152*a^3*b^2*c^4*d*
(c + d*x)^2)/(a + b*x)^2 + (1152*a^4*b*c^3*d^2*(c + d*x)^2)/(a + b*x)^2 + (896*a^5*c^2*d^3*(c + d*x)^2)/(a + b
*x)^2 + (790*a^3*b^3*c^4*(c + d*x)^3)/(a + b*x)^3 - (630*a^4*b^2*c^3*d*(c + d*x)^3)/(a + b*x)^3 - (630*a^5*b*c
^2*d^2*(c + d*x)^3)/(a + b*x)^3 - (490*a^6*c*d^3*(c + d*x)^3)/(a + b*x)^3 + (105*a^4*b^3*c^3*(c + d*x)^4)/(a +
b*x)^4 + (135*a^5*b^2*c^2*d*(c + d*x)^4)/(a + b*x)^4 + (135*a^6*b*c*d^2*(c + d*x)^4)/(a + b*x)^4 + (105*a^7*d
^3*(c + d*x)^4)/(a + b*x)^4))/(1920*a^4*c^4*Sqrt[a + b*x]*(-c + (a*(c + d*x))/(a + b*x))^5) - ((-(b*c) + a*d)^
2*(7*b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + 7*a^3*d^3)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*
x])])/(128*a^(9/2)*c^(9/2))

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fricas [A]  time = 17.26, size = 730, normalized size = 2.12 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d - 34 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 11 \, a^{3} b^{2} c^{4} d - 11 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{5} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d - 34 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 11 \, a^{3} b^{2} c^{4} d - 11 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} c^{5} x^{5}}\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^6,x, algorithm="fricas")

[Out]

[1/7680*(15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*sq
rt(a*c)*x^5*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*(384*a^5*c^5 - (105*a*b^4*c^5 - 40*a^2*b^3*c^4*d - 34
*a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 105*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 11*a^3*b^2*c^4*d - 11*a^4*b*c^3
*d^2 + 35*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 2*a^4*b*c^4*d + 7*a^5*c^3*d^2)*x^2 + 48*(a^4*b*c^5 + a^5*c^4*d
)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^5*x^5), 1/3840*(15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2
*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqr
t(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*(384*a^5*c^5 - (105*a*b^4*c^5 -
40*a^2*b^3*c^4*d - 34*a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 105*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 11*a^3*b^2
*c^4*d - 11*a^4*b*c^3*d^2 + 35*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 2*a^4*b*c^4*d + 7*a^5*c^3*d^2)*x^2 + 48*(
a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^5*x^5)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^6,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.02, size = 967, normalized size = 2.80 \begin {gather*} -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 a^{5} d^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-75 a^{4} b c \,d^{4} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-30 a^{3} b^{2} c^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-30 a^{2} b^{3} c^{3} d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-75 a \,b^{4} c^{4} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+105 b^{5} c^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-210 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}+68 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+80 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}-210 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-44 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{2} d^{2} x^{3}-44 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{3} d \,x^{3}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}+32 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x +96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x +768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^4*(105*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2))/x)*x^5*a^5*d^5-75*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*a^4*b*c*
d^4-30*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*a^3*b^2*c^2*d^3-30*ln((a*d*
x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*a^2*b^3*c^3*d^2-75*ln((a*d*x+b*c*x+2*a*c+2
*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*a*b^4*c^4*d+105*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*b^5*c^5-210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^4*d^4+80*(a*c
)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b*c*d^3+68*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2
*b^2*c^2*d^2+80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^3*c^3*d-210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^4*b^4*c^4+140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^4*c*d^3-44*(a*c)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b*c^2*d^2-44*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^2*c^3*d+14
0*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^3*c^4-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^
2*a^4*c^2*d^2+32*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b*c^3*d-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*x^2*a^2*b^2*c^4+96*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c^3*d+96*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*b*c^4+768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^4*(a*c)^(1/2))/(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)/x^5/(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^6,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 [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^6,x)

[Out]

\text{Hanged}

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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^{6}}\, 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**6,x)

[Out]

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**6, x)

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