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

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

[Out]

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

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Rubi [A]  time = 0.365873, 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} \[ -\frac{b \sqrt{c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt{a+b x}}+\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{11/2} \sqrt{c}}+\frac{3 c \sqrt{c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}-\frac{7 b \sqrt{c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac{\sqrt{c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.310447, size = 199, normalized size = 0.72 \[ \frac{-x^2 \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \left (3 a^{5/2} (c+d x)^{5/2}+5 x (b c-a d) \left (\sqrt{a} \sqrt{c+d x} (4 a c+a d x+3 b c x)-3 c^{3/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )\right )+2 a^{7/2} x (c+d x)^{7/2} (9 b c-a d)-8 a^{9/2} c (c+d x)^{7/2}}{24 a^{11/2} c^2 x^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.03, 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((d*x+c)^(5/2)/x^4/(b*x+a)^(5/2),x)

[Out]

-1/48*(d*x+c)^(1/2)*(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*a^3*b^2*d^3-225*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^2*b^3*c*d^2+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*b^4*c^2*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^5*b^5*c^3+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*a^4*b*d^3-450*l
n((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*b^2*c*d^2+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^3*c^2*d-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*b^4*c^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^3*a^5*d^3-2
25*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^4*b*c*d^2+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^3*b^2*c^2*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^2*b^3*c^3+226*x^4*a^2*b^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-840*x^4*a*b^3*c*d*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+630*x^4*b^4*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+324*x^3*a^3*b*d^2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)-1148*x^3*a^2*b^2*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+840*x^3*a*b^3*c^2*(a*c)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)+66*x^2*a^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-192*x^2*a^3*b*c*d*(a*c)^(1/2)
*((b*x+a)*(d*x+c))^(1/2)+126*x^2*a^2*b^2*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+52*x*a^4*c*d*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)-36*x*a^3*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+16*a^4*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c)
)^(1/2))/a^5/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^3/(b*x+a)^(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((d*x+c)^(5/2)/x^4/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 39.2591, size = 1829, 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((d*x+c)^(5/2)/x^4/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: TypeError