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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.396512, antiderivative size = 346, normalized size of antiderivative = 1., 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 = {98, 149, 151, 12, 93, 208} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (481 a^2 b c d^2-15 a^3 d^3-749 a b^2 c^2 d+315 b^3 c^3\right )}{960 a^4 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{240 a^3 x^3}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4-2310 a b^3 c^3 d+945 b^4 c^4\right )}{1920 a^5 c^2 x}+\frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{11/2} c^{5/2}}+\frac{c \sqrt{a+b x} \sqrt{c+d x} (9 b c-13 a d)}{40 a^2 x^4}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{5 a x^5} \]

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 98

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 149

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] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 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 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{(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 ) \operatorname{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 = 0.288746, size = 232, normalized size = 0.67 \[ \frac{\frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac{5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}-8 \sqrt{a+b x} (c+d x)^{5/2}\right )}{24 a x^3}+\frac{6 \sqrt{a+b x} (c+d x)^{7/2} (a d+3 b c)}{x^4}-\frac{16 a c \sqrt{a+b x} (c+d x)^{7/2}}{x^5}}{80 a^2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-16*a*c*Sqrt[a + b*x]*(c + d*x)^(7/2))/x^5 + (6*(3*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/x^4 + ((63*b^2*
c^2 + 14*a*b*c*d + 3*a^2*d^2)*(-8*Sqrt[a + b*x]*(c + d*x)^(5/2) + (5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c]*Sqrt[a + b
*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])]))/(a^(5/2)*Sqrt[c])))/(24*a*x^3))/(80*a^2*c^2)

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Maple [B]  time = 0.026, size = 813, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

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)

[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)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x
)*x^5*a^5*d^5+75*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^4*b*c*d^4+450*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^3*b^2*c^2*d^3-2250*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^2*b^3*c^3*d^2+2625*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)+2*a*c)/x)*x^5*a*b^4*c^4*d-945*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*b^5*c^5-9
0*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^4*a^4*d^4-180*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^4*a^3*b*c*d^3+3128
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^4*a^2*b^2*c^2*d^2-4620*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^4*a*b^3*c^
3*d+1890*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^4*b^4*c^4+60*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^4*c*d^3-
1924*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^3*b*c^2*d^2+2996*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^2*b^
2*c^3*d-1260*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a*b^3*c^4+1488*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^
4*c^2*d^2-2368*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^3*b*c^3*d+1008*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^
2*a^2*b^2*c^4+2016*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^4*c^3*d-864*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a
^3*b*c^4+768*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*c^4)/((b*x+a)*(d*x+c))^(1/2)/x^5/(a*c)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 60.9988, size = 1643, normalized size = 4.75 \begin{align*} \text{result too large to display} \end{align*}

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)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError