3.8.22 \(\int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx\) [722]

Optimal. Leaf size=346 \[ \frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}} \]

[Out]

1/128*(-a*d+b*c)^3*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(1
1/2)/c^(5/2)-1/5*c*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a/x^5+1/40*c*(-13*a*d+9*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/x^
4-1/240*(93*a^2*d^2-148*a*b*c*d+63*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/x^3+1/960*(-15*a^3*d^3+481*a^2*b*c
*d^2-749*a*b^2*c^2*d+315*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c/x^2-1/1920*(-45*a^4*d^4-90*a^3*b*c*d^3+156
4*a^2*b^2*c^2*d^2-2310*a*b^3*c^3*d+945*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^5/c^2/x

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 346, 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 = {100, 154, 156, 12, 95, 214} \begin {gather*} \frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{40 a^2 x^4}+\frac {\left (3 a^2 d^2+14 a b c d+63 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^{11/2} c^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{240 a^3 x^3}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{960 a^4 c x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(x^6*Sqrt[a + b*x]),x]

[Out]

(c*(9*b*c - 13*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a^2*x^4) - ((63*b^2*c^2 - 148*a*b*c*d + 93*a^2*d^2)*Sqrt[
a + b*x]*Sqrt[c + d*x])/(240*a^3*x^3) + ((315*b^3*c^3 - 749*a*b^2*c^2*d + 481*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a
+ b*x]*Sqrt[c + d*x])/(960*a^4*c*x^2) - ((945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 90*a^3*b*c*
d^3 - 45*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^5*c^2*x) - (c*Sqrt[a + b*x]*(c + d*x)^(3/2))/(5*a*x^5)
+ ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])
])/(128*a^(11/2)*c^(5/2))

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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx &=-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (9 b c-13 a d)+d (3 b c-5 a d) x\right )}{x^5 \sqrt {a+b x}} \, dx}{5 a}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\int \frac {-\frac {1}{4} c \left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right )-\frac {1}{2} d \left (27 b^2 c^2-63 a b c d+40 a^2 d^2\right ) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 a^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {\int \frac {-\frac {1}{8} c \left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right )-\frac {1}{2} b c d \left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a^3 c}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\int \frac {-\frac {1}{16} c \left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right )-\frac {1}{8} b c d \left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^4 c^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {\int -\frac {15 c (b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^5 c^3}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^5 c^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 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^5 c^2}\\ &=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.59, size = 258, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (945 b^4 c^4 x^4-210 a b^3 c^3 x^3 (3 c+11 d x)+2 a^2 b^2 c^2 x^2 \left (252 c^2+749 c d x+782 d^2 x^2\right )-2 a^3 b c x \left (216 c^3+592 c^2 d x+481 c d^2 x^2+45 d^3 x^3\right )+3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )\right )}{1920 a^5 c^2 x^5}+\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(x^6*Sqrt[a + b*x]),x]

[Out]

-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(945*b^4*c^4*x^4 - 210*a*b^3*c^3*x^3*(3*c + 11*d*x) + 2*a^2*b^2*c^2*x^2*(
252*c^2 + 749*c*d*x + 782*d^2*x^2) - 2*a^3*b*c*x*(216*c^3 + 592*c^2*d*x + 481*c*d^2*x^2 + 45*d^3*x^3) + 3*a^4*
(128*c^4 + 336*c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)))/(a^5*c^2*x^5) + ((b*c - a*d)^3*(63*b^2
*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(11/2)*c^(5/2)
)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(812\) vs. \(2(302)=604\).
time = 0.07, size = 813, normalized size = 2.35

method	result	size
default	\(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (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^{5} d^{5} x^{5}+75 \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}+450 \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}-2250 \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}+2625 \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}-945 \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}-90 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4} x^{4}-180 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3} x^{4}+3128 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}-4620 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d \,x^{4}+1890 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4} x^{4}+60 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c \,d^{3} x^{3}-1924 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{2} d^{2} x^{3}+2996 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{3} d \,x^{3}-1260 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{2} d^{2} x^{2}-2368 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{3} d \,x^{2}+1008 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{3} d x -864 \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^{5} c^{2} \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((d*x+c)^(5/2)/x^6/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3840*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^5/c^2*(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^5*d^5*x^5+75*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+450*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^5-2250*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+2625*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^5-945*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^5-9
0*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*d^4*x^4-180*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c*d^3*x^4+3128
*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^2*d^2*x^4-4620*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^3*d*
x^4+1890*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^4*c^4*x^4+60*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c*d^3*x^3-
1924*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^2*d^2*x^3+2996*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^
3*d*x^3-1260*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^4*x^3+1488*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^
2*d^2*x^2-2368*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^3*d*x^2+1008*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^
2*b^2*c^4*x^2+2016*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^3*d*x-864*(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((d*x+c)^(5/2)/x^6/(b*x+a)^(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 detail

________________________________________________________________________________________

Fricas [A]
time = 5.67, size = 732, normalized size = 2.12 \begin {gather*} \left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, 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 (945 \, a b^{4} c^{5} - 2310 \, a^{2} b^{3} c^{4} d + 1564 \, a^{3} b^{2} c^{3} d^{2} - 90 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (315 \, a^{2} b^{3} c^{5} - 749 \, a^{3} b^{2} c^{4} d + 481 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (63 \, a^{3} b^{2} c^{5} - 148 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} - 144 \, {\left (3 \, a^{4} b c^{5} - 7 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{6} c^{3} x^{5}}, -\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, 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 (945 \, a b^{4} c^{5} - 2310 \, a^{2} b^{3} c^{4} d + 1564 \, a^{3} b^{2} c^{3} d^{2} - 90 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (315 \, a^{2} b^{3} c^{5} - 749 \, a^{3} b^{2} c^{4} d + 481 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (63 \, a^{3} b^{2} c^{5} - 148 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} - 144 \, {\left (3 \, a^{4} b c^{5} - 7 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{6} c^{3} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^6/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*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 + (945*a*b^4*c^5 - 2310*a^2*b^3*c
^4*d + 1564*a^3*b^2*c^3*d^2 - 90*a^4*b*c^2*d^3 - 45*a^5*c*d^4)*x^4 - 2*(315*a^2*b^3*c^5 - 749*a^3*b^2*c^4*d +
481*a^4*b*c^3*d^2 - 15*a^5*c^2*d^3)*x^3 + 8*(63*a^3*b^2*c^5 - 148*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 - 144*(3*a
^4*b*c^5 - 7*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*c^3*x^5), -1/3840*(15*(63*b^5*c^5 - 175*a*b^4*c^4
*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*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*(38
4*a^5*c^5 + (945*a*b^4*c^5 - 2310*a^2*b^3*c^4*d + 1564*a^3*b^2*c^3*d^2 - 90*a^4*b*c^2*d^3 - 45*a^5*c*d^4)*x^4
- 2*(315*a^2*b^3*c^5 - 749*a^3*b^2*c^4*d + 481*a^4*b*c^3*d^2 - 15*a^5*c^2*d^3)*x^3 + 8*(63*a^3*b^2*c^5 - 148*a
^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 - 144*(3*a^4*b*c^5 - 7*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*c^3*x^
5)]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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((d*x+c)**(5/2)/x**6/(b*x+a)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5928 vs. \(2 (302) = 604\).
time = 37.52, size = 5928, normalized size = 17.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^6/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/1920*(15*(63*sqrt(b*d)*b^6*c^5*abs(b) - 175*sqrt(b*d)*a*b^5*c^4*d*abs(b) + 150*sqrt(b*d)*a^2*b^4*c^3*d^2*abs
(b) - 30*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) - 5*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) - 3*sqrt(b*d)*a^5*b*d^5*abs(b))*a
rctan(-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^5*b*c^2) - 2*(945*sqrt(b*d)*b^24*c^14*abs(b) - 11760*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 6
7189*sqrt(b*d)*a^2*b^22*c^12*d^2*abs(b) - 233080*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) + 546885*sqrt(b*d)*a^4*b^2
0*c^10*d^4*abs(b) - 914520*sqrt(b*d)*a^5*b^19*c^9*d^5*abs(b) + 1117785*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) - 100
6128*sqrt(b*d)*a^7*b^17*c^7*d^7*abs(b) + 661395*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) - 308640*sqrt(b*d)*a^9*b^15*
c^5*d^9*abs(b) + 95775*sqrt(b*d)*a^10*b^14*c^4*d^10*abs(b) - 16600*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) + 439*s
qrt(b*d)*a^12*b^12*c^2*d^12*abs(b) + 360*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) - 45*sqrt(b*d)*a^14*b^10*d^14*abs(b
) - 8505*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) + 79065*
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) - 316630*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) + 692110*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) - 818075*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) + 269155*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) + 707260*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) - 1290380*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) + 1092385*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) - 535025*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) + 144250*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^12*c^3*d^10*abs(b) - 14050*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^11*b^11*c^2*d^11*abs(b) - 1965*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^12*b^10*c*d^12*abs(b) + 405*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^13*b^9*d^13*abs(b) + 34020*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) - 226800*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) + 600960*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) - 762800*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) + 437700*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) - 263520*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) + 820000*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) - 1388640*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) + 1142700*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) - 474800*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) + 79200*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) + 3600*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) - 1620*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) - 79380*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) + 361620*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) - 592340*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) + 396660*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) - 75240*sqrt(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) - 192280*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) + 752120*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) - 1139640*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) + 763260*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^10*c^3*d^8*abs(b) - 198140*sqrt(b*d)*(sqrt(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) - 420*sqrt(b*...

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^6\,\sqrt {a+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/2)/(x^6*(a + b*x)^(1/2)),x)

[Out]

int((c + d*x)^(5/2)/(x^6*(a + b*x)^(1/2)), x)

________________________________________________________________________________________