3.697 \(\int \frac{(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=278 \[ -\frac{d \sqrt{a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{24 c^5 \sqrt{c+d x}}-\frac{5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{11/2}}-\frac{3 a \sqrt{a+b x} (b c-a d)}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{7 d \sqrt{a+b x} (7 b c-15 a d) (b c-a d)}{24 c^4 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (11 b c-21 a d) (b c-a d)}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}} \]

[Out]

(-7*d*(7*b*c - 15*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(24*c^4*(c + d*x)^(3/2)) - (3*a*(b*c - a*d)*Sqrt[a + b*x])/(
4*c^2*x^2*(c + d*x)^(3/2)) - ((11*b*c - 21*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(8*c^3*x*(c + d*x)^(3/2)) - (a*(a +
b*x)^(3/2))/(3*c*x^3*(c + d*x)^(3/2)) - (d*(113*b^2*c^2 - 420*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])/(24*c^5*S
qrt[c + d*x]) - (5*(b*c - a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sq
rt[c + d*x])])/(8*Sqrt[a]*c^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.346778, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {98, 149, 151, 152, 12, 93, 208} \[ -\frac{d \sqrt{a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{24 c^5 \sqrt{c+d x}}-\frac{5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{11/2}}-\frac{3 a \sqrt{a+b x} (b c-a d)}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{7 d \sqrt{a+b x} (7 b c-15 a d) (b c-a d)}{24 c^4 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (11 b c-21 a d) (b c-a d)}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)/(x^4*(c + d*x)^(5/2)),x]

[Out]

(-7*d*(7*b*c - 15*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(24*c^4*(c + d*x)^(3/2)) - (3*a*(b*c - a*d)*Sqrt[a + b*x])/(
4*c^2*x^2*(c + d*x)^(3/2)) - ((11*b*c - 21*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(8*c^3*x*(c + d*x)^(3/2)) - (a*(a +
b*x)^(3/2))/(3*c*x^3*(c + d*x)^(3/2)) - (d*(113*b^2*c^2 - 420*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])/(24*c^5*S
qrt[c + d*x]) - (5*(b*c - a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sq
rt[c + d*x])])/(8*Sqrt[a]*c^(11/2))

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 152

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] && IntegersQ[2*m, 2*n, 2*p]

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{(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx &=-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac{\int \frac{\sqrt{a+b x} \left (-\frac{9}{2} a (b c-a d)-3 b (b c-a d) x\right )}{x^3 (c+d x)^{5/2}} \, dx}{3 c}\\ &=-\frac{3 a (b c-a d) \sqrt{a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac{\int \frac{-\frac{3}{4} a (11 b c-21 a d) (b c-a d)-\frac{3}{2} b (4 b c-9 a d) (b c-a d) x}{x^2 \sqrt{a+b x} (c+d x)^{5/2}} \, dx}{6 c^2}\\ &=-\frac{3 a (b c-a d) \sqrt{a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{(11 b c-21 a d) (b c-a d) \sqrt{a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}+\frac{\int \frac{\frac{15}{8} a (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )-\frac{3}{2} a b d (11 b c-21 a d) (b c-a d) x}{x \sqrt{a+b x} (c+d x)^{5/2}} \, dx}{6 a c^3}\\ &=-\frac{7 d (7 b c-15 a d) (b c-a d) \sqrt{a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac{3 a (b c-a d) \sqrt{a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{(11 b c-21 a d) (b c-a d) \sqrt{a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac{\int \frac{-\frac{45}{16} a (b c-a d)^2 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )+\frac{21}{8} a b d (7 b c-15 a d) (b c-a d)^2 x}{x \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{9 a c^4 (b c-a d)}\\ &=-\frac{7 d (7 b c-15 a d) (b c-a d) \sqrt{a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac{3 a (b c-a d) \sqrt{a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{(11 b c-21 a d) (b c-a d) \sqrt{a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac{d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt{a+b x}}{24 c^5 \sqrt{c+d x}}+\frac{2 \int \frac{45 a (b c-a d)^3 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )}{32 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{9 a c^5 (b c-a d)^2}\\ &=-\frac{7 d (7 b c-15 a d) (b c-a d) \sqrt{a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac{3 a (b c-a d) \sqrt{a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{(11 b c-21 a d) (b c-a d) \sqrt{a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac{d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt{a+b x}}{24 c^5 \sqrt{c+d x}}+\frac{\left (5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 c^5}\\ &=-\frac{7 d (7 b c-15 a d) (b c-a d) \sqrt{a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac{3 a (b c-a d) \sqrt{a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{(11 b c-21 a d) (b c-a d) \sqrt{a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac{d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt{a+b x}}{24 c^5 \sqrt{c+d x}}+\frac{\left (5 (b c-a d) \left (b^2 c^2-14 a b c d+21 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 )}{8 c^5}\\ &=-\frac{7 d (7 b c-15 a d) (b c-a d) \sqrt{a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac{3 a (b c-a d) \sqrt{a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{(11 b c-21 a d) (b c-a d) \sqrt{a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac{d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt{a+b x}}{24 c^5 \sqrt{c+d x}}-\frac{5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.318875, size = 198, normalized size = 0.71 \[ \frac{-x^2 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (3 c^{5/2} (a+b x)^{5/2}-5 x (b c-a d) \left (\sqrt{c} \sqrt{a+b x} (4 a c+3 a d x+b c x)-3 a^{3/2} (c+d x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )\right )-2 c^{7/2} x (a+b x)^{7/2} (b c-9 a d)-8 a c^{9/2} (a+b x)^{7/2}}{24 a^2 c^{11/2} x^3 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)/(x^4*(c + d*x)^(5/2)),x]

[Out]

(-8*a*c^(9/2)*(a + b*x)^(7/2) - 2*c^(7/2)*(b*c - 9*a*d)*x*(a + b*x)^(7/2) - (b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2
)*x^2*(3*c^(5/2)*(a + b*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[c]*Sqrt[a + b*x]*(4*a*c + b*c*x + 3*a*d*x) - 3*a^(3/2
)*(c + d*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))/(24*a^2*c^(11/2)*x^3*(c + d*x)^(
3/2))

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Maple [B]  time = 0.031, size = 1009, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)/x^4/(d*x+c)^(5/2),x)

[Out]

1/48*(b*x+a)^(1/2)*(315*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*d^5-525*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*c*d^4+225*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^2*c^2*d^3-15*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^3*c^3*d^2+630*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^4*a^3*c*d^4-1050*
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^4*a^2*b*c^2*d^3+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^4*a*b^2*c^3*d^2-30*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^4*b^3*c^4*d+315*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^3*a^3*c^2*d
^3-525*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^3*a^2*b*c^3*d^2+225*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^3*a*b^2*c^4*d-15*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^3*b^3*c^5-630*x^4*a^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+840*x^4*a*b*c*d^3*((b*x+a)
*(d*x+c))^(1/2)*(a*c)^(1/2)-226*x^4*b^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-840*x^3*a^2*c*d^3*((b*x+a)
*(d*x+c))^(1/2)*(a*c)^(1/2)+1148*x^3*a*b*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-324*x^3*b^2*c^3*d*((b*x+a
)*(d*x+c))^(1/2)*(a*c)^(1/2)-126*x^2*a^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+192*x^2*a*b*c^3*d*((b*x+a
)*(d*x+c))^(1/2)*(a*c)^(1/2)-66*x^2*b^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+36*x*a^2*c^3*d*((b*x+a)*(d*x+c
))^(1/2)*(a*c)^(1/2)-52*x*a*b*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-16*a^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)
^(1/2))/c^5/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^3/(d*x+c)^(3/2)

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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((b*x+a)^(5/2)/x^4/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 52.5179, size = 1828, normalized size = 6.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)/x^4/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/96*(15*((b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 35*a^2*b*c*d^4 - 21*a^3*d^5)*x^5 + 2*(b^3*c^4*d - 15*a*b^2*c^3*d
^2 + 35*a^2*b*c^2*d^3 - 21*a^3*c*d^4)*x^4 + (b^3*c^5 - 15*a*b^2*c^4*d + 35*a^2*b*c^3*d^2 - 21*a^3*c^2*d^3)*x^3
)*sqrt(a*c)*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*(8*a^3*c^5 + (113*a*b^2*c^3*d^2 - 420*a^2*b*c^2*d^3 +
315*a^3*c*d^4)*x^4 + 2*(81*a*b^2*c^4*d - 287*a^2*b*c^3*d^2 + 210*a^3*c^2*d^3)*x^3 + 3*(11*a*b^2*c^5 - 32*a^2*
b*c^4*d + 21*a^3*c^3*d^2)*x^2 + 2*(13*a^2*b*c^5 - 9*a^3*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^6*d^2*x^5
+ 2*a*c^7*d*x^4 + a*c^8*x^3), 1/48*(15*((b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 35*a^2*b*c*d^4 - 21*a^3*d^5)*x^5 + 2
*(b^3*c^4*d - 15*a*b^2*c^3*d^2 + 35*a^2*b*c^2*d^3 - 21*a^3*c*d^4)*x^4 + (b^3*c^5 - 15*a*b^2*c^4*d + 35*a^2*b*c
^3*d^2 - 21*a^3*c^2*d^3)*x^3)*sqrt(-a*c)*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*(8*a^3*c^5 + (113*a*b^2*c^3*d^2 - 420*a^2*b*c^2*d^3
+ 315*a^3*c*d^4)*x^4 + 2*(81*a*b^2*c^4*d - 287*a^2*b*c^3*d^2 + 210*a^3*c^2*d^3)*x^3 + 3*(11*a*b^2*c^5 - 32*a^2
*b*c^4*d + 21*a^3*c^3*d^2)*x^2 + 2*(13*a^2*b*c^5 - 9*a^3*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^6*d^2*x^5
+ 2*a*c^7*d*x^4 + a*c^8*x^3)]

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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((b*x+a)**(5/2)/x**4/(d*x+c)**(5/2),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((b*x+a)^(5/2)/x^4/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError