3.7.6 \(\int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx\) [606]

Optimal. Leaf size=340 \[ -\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{9/2}} \]

[Out]

1/128*(-a*d+b*c)^3*(7*a^2*d^2+6*a*b*c*d+3*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^(9/2)-1/5*(b*x+a)^(3/2)*(d*x+c)^(1/2)/x^5-1/40*(a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c/x^4-1/240*(3*b^2*
c/a+12*b*d-7*a*d^2/c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c/x^3+1/960*(-35*a^3*d^3+61*a^2*b*c*d^2-9*a*b^2*c^2*d+15*b^3
*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^3/x^2-1/1920*(-105*a^4*d^4+190*a^3*b*c*d^3-36*a^2*b^2*c^2*d^2-30*a*b^3
*c^3*d+45*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4/x

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {99, 154, 156, 12, 95, 214} \begin {gather*} \frac {\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{9/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{960 a^2 c^3 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{1920 a^3 c^4 x}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{240 c x^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{40 c x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]

[Out]

-1/40*((3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(c*x^4) - (((3*b^2*c)/a + 12*b*d - (7*a*d^2)/c)*Sqrt[a + b*x
]*Sqrt[c + d*x])/(240*c*x^3) + ((15*b^3*c^3 - 9*a*b^2*c^2*d + 61*a^2*b*c*d^2 - 35*a^3*d^3)*Sqrt[a + b*x]*Sqrt[
c + d*x])/(960*a^2*c^3*x^2) - ((45*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4*d
^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^3*c^4*x) - ((a + b*x)^(3/2)*Sqrt[c + d*x])/(5*x^5) + ((b*c - a*d)^3*(
3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*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 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 154

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

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 {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx &=-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {1}{5} \int \frac {\sqrt {a+b x} \left (\frac {1}{2} (3 b c+a d)+2 b d x\right )}{x^5 \sqrt {c+d x}} \, dx\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {\int \frac {\frac {1}{4} \left (3 b^2 c^2+12 a b c d-7 a^2 d^2\right )+\frac {1}{2} b d (7 b c-3 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 c}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\int \frac {\frac {1}{8} \left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right )+\frac {1}{2} b d \left (3 b^2 c^2+12 a b c d-7 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a c^2}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {\int \frac {\frac {1}{16} \left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right )+\frac {1}{8} b d \left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^2 c^3}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\int \frac {15 (b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^3 c^4}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 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^3 c^4}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 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 )}{128 a^3 c^4}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {(b c-a d)^3 \left (3 b^2 c^2+6 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^{7/2} c^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.56, size = 254, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (45 b^4 c^4 x^4-30 a b^3 c^3 x^3 (c+d x)+6 a^2 b^2 c^2 x^2 \left (4 c^2+3 c d x-6 d^2 x^2\right )+2 a^3 b c x \left (264 c^3+48 c^2 d x-61 c d^2 x^2+95 d^3 x^3\right )+a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )\right )}{1920 a^3 c^4 x^5}+\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{7/2} c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]

[Out]

-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(45*b^4*c^4*x^4 - 30*a*b^3*c^3*x^3*(c + d*x) + 6*a^2*b^2*c^2*x^2*(4*c^2 +
3*c*d*x - 6*d^2*x^2) + 2*a^3*b*c*x*(264*c^3 + 48*c^2*d*x - 61*c*d^2*x^2 + 95*d^3*x^3) + a^4*(384*c^4 + 48*c^3
*d*x - 56*c^2*d^2*x^2 + 70*c*d^3*x^3 - 105*d^4*x^4)))/(a^3*c^4*x^5) + ((b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d +
7*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(7/2)*c^(9/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(812\) vs. \(2(296)=592\).
time = 0.06, size = 813, normalized size = 2.39

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 \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^{5} d^{5} x^{5}-225 \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} b c \,d^{4} x^{5}+90 \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^{2} c^{2} d^{3} x^{5}+30 \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^{3} c^{3} d^{2} x^{5}+45 \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^{4} c^{4} d \,x^{5}-45 \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^{5} c^{5} x^{5}-210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4} x^{4}+380 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3} x^{4}-72 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}-60 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d \,x^{4}+90 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4} x^{4}+140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c \,d^{3} x^{3}-244 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{2} d^{2} x^{3}+36 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{3} d \,x^{3}-60 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{2} d^{2} x^{2}+192 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{3} d \,x^{2}+48 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{3} d x +1056 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{4} x +768 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{3} c^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{5} \sqrt {a c}}\)	\(813\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(105*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^5*d^5*x^5-225*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*b*c*d^4*x^5+90*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^2*c^2*d^3*x^5+30*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^3*c^3*d^2*x^5+45*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^4*c^4*d*x^5-45*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^5*c^5*x^5-210*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*d^4*x^4+380*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c*d^3*x^4-72*(a*c
)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^2*d^2*x^4-60*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^3*d*x^4+90*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^4*c^4*x^4+140*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c*d^3*x^3-244*(a*c
)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^2*d^2*x^3+36*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^3*d*x^3-60*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^4*x^3-112*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^2*d^2*x^2+192*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^3*d*x^2+48*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^4*x^2+96*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^3*d*x+1056*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^4*x+768*((d*x
+c)*(b*x+a))^(1/2)*a^4*c^4*(a*c)^(1/2))/((d*x+c)*(b*x+a))^(1/2)/x^5/(a*c)^(1/2)

________________________________________________________________________________________

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)^(3/2)*(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 detail

________________________________________________________________________________________

Fricas [A]
time = 6.11, size = 730, normalized size = 2.15 \begin {gather*} \left [-\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, 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 (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{4} c^{5} x^{5}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, 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 (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{4} c^{5} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[-1/7680*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*a^5*d^5)*
sqrt(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 + (45*a*b^4*c^5 - 30*a^2*b^3*c^4*d - 3
6*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 105*a^5*c*d^4)*x^4 - 2*(15*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 61*a^4*b*c^
3*d^2 - 35*a^5*c^2*d^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^2)*x^2 + 48*(11*a^4*b*c^5 + a^5*
c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^5), -1/3840*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d
^2 - 6*a^3*b^2*c^2*d^3 + 15*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)*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*(384*a^5*c^5 + (45*a*b^4*
c^5 - 30*a^2*b^3*c^4*d - 36*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 105*a^5*c*d^4)*x^4 - 2*(15*a^2*b^3*c^5 - 9*a
^3*b^2*c^4*d + 61*a^4*b*c^3*d^2 - 35*a^5*c^2*d^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^2)*x^2
+ 48*(11*a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^5)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(d*x+c)**(1/2)/x**6,x)

[Out]

Integral((a + b*x)**(3/2)*sqrt(c + d*x)/x**6, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5872 vs. \(2 (296) = 592\).
time = 37.73, size = 5872, normalized size = 17.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x, algorithm="giac")

[Out]

1/1920*(15*(3*sqrt(b*d)*b^6*c^5*abs(b) - 3*sqrt(b*d)*a*b^5*c^4*d*abs(b) - 2*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) -
6*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) + 15*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) - 7*sqrt(b*d)*a^5*b*d^5*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^4) - 2*(45*sqrt(b*d)*b^24*c^14*abs(b) - 480*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 2289*sqrt
(b*d)*a^2*b^22*c^12*d^2*abs(b) - 6200*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) + 9425*sqrt(b*d)*a^4*b^20*c^10*d^4*ab
s(b) - 3720*sqrt(b*d)*a^5*b^19*c^9*d^5*abs(b) - 18075*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) + 49872*sqrt(b*d)*a^7*
b^17*c^7*d^7*abs(b) - 71865*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) + 68880*sqrt(b*d)*a^9*b^15*c^5*d^9*abs(b) - 4612
5*sqrt(b*d)*a^10*b^14*c^4*d^10*abs(b) + 21480*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) - 6661*sqrt(b*d)*a^12*b^12*c
^2*d^12*abs(b) + 1240*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) - 105*sqrt(b*d)*a^14*b^10*d^14*abs(b) - 405*sqrt(b*d)*
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^22*c^13*abs(b) + 3045*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^21*c^12*d*abs(b) - 9150*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^20*c^11*d^2*abs(b) + 10550*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^19*c^10*d^3*abs(b) + 12625*sqrt(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^18*c^9*d^4*abs(b) - 65945*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^17*c^8*d^5*abs(b) + 109900*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^16*c^7*d^6*abs(b) - 90140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^15*c^6*d^7*abs(b) + 13405*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^14*c^5*d^8*abs(b) + 50275*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^13*c^4*d^9*abs(b) - 56750*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^12*c^3*d^10*abs(b) + 29830*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^11*b^11*c^2*d^11*abs(b) - 8185*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^12*b^10*c*d^12*abs(b) + 945*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^2*a^13*b^9*d^13*abs(b) + 1620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4*b^20*c^12*abs(b) - 7920*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^19*c^11*d*abs(b) + 12480*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^18*c^10*d^2*abs(b) + 6480*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^17*c^9*d^3*abs(b) - 55820*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^16*c^8*d^4*abs(b) + 89760*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^15*c^7*d^5*abs(b) - 58080*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^14*c^6*d^6*abs(b) + 4000*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^13*c^5*d^7*abs(b) - 8580*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^12*c^4*d^8*abs(b) + 48720*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
^11*c^3*d^9*abs(b) - 51680*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^10*b^
10*c^2*d^10*abs(b) + 22800*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^11*b^
9*c*d^11*abs(b) - 3780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^12*b^8*d^
12*abs(b) - 3780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^18*c^11*abs(b)
+ 10500*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^17*c^10*d*abs(b) - 474
0*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^16*c^9*d^2*abs(b) - 11420*
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^15*c^8*d^3*abs(b) + 3000*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^14*c^7*d^4*abs(b) + 33480*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^13*c^6*d^5*abs(b) - 39400*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^12*c^5*d^6*abs(b) + 8360*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^11*c^4*d^7*abs(b) - 12660*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^10*c^3*d^8*abs(b) + 42420*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^9*b^9*c^2*d^9*abs(b) - 34580*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d ...

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(3/2)*(c + d*x)^(1/2))/x^6,x)

[Out]

int(((a + b*x)^(3/2)*(c + d*x)^(1/2))/x^6, x)

________________________________________________________________________________________