3.575 \(\int \frac{\sqrt{a+b x} (c+d x)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=436 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4-308 a b^3 c^3 d+105 b^4 c^4\right )}{3840 a^4 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (51 a^2 b c d^2+5 a^3 d^3-61 a b^2 c^2 d+21 b^3 c^3\right )}{960 a^3 c x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{160 a^2 x^4}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5-945 a b^4 c^4 d+315 b^5 c^5\right )}{7680 a^5 c^3 x}+\frac{\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{11/2} c^{7/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{60 a x^5} \]

[Out]

((3*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(160*a^2*x^4) - ((21*b^3*c^3 - 61*a*b^2*c^2*
d + 51*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^3*c*x^3) + ((105*b^4*c^4 - 308*a*b^3*c^3*d
+ 262*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 25*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(3840*a^4*c^2*x^2) - ((315*
b^5*c^5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 75*a^5*d^5)*Sqrt[a + b
*x]*Sqrt[c + d*x])/(7680*a^5*c^3*x) - ((b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(60*a*x^5) - (Sqrt[a + b*x
]*(c + d*x)^(5/2))/(6*x^6) + ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*
x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(11/2)*c^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.46887, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {97, 149, 151, 12, 93, 208} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4-308 a b^3 c^3 d+105 b^4 c^4\right )}{3840 a^4 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (51 a^2 b c d^2+5 a^3 d^3-61 a b^2 c^2 d+21 b^3 c^3\right )}{960 a^3 c x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{160 a^2 x^4}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5-945 a b^4 c^4 d+315 b^5 c^5\right )}{7680 a^5 c^3 x}+\frac{\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{11/2} c^{7/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{60 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]

[Out]

((3*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(160*a^2*x^4) - ((21*b^3*c^3 - 61*a*b^2*c^2*
d + 51*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^3*c*x^3) + ((105*b^4*c^4 - 308*a*b^3*c^3*d
+ 262*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 25*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(3840*a^4*c^2*x^2) - ((315*
b^5*c^5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 75*a^5*d^5)*Sqrt[a + b
*x]*Sqrt[c + d*x])/(7680*a^5*c^3*x) - ((b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(60*a*x^5) - (Sqrt[a + b*x
]*(c + d*x)^(5/2))/(6*x^6) + ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*
x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(11/2)*c^(7/2))

Rule 97

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 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{\sqrt{a+b x} (c+d x)^{5/2}}{x^7} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}+\frac{1}{6} \int \frac{(c+d x)^{3/2} \left (\frac{1}{2} (b c+5 a d)+3 b d x\right )}{x^6 \sqrt{a+b x}} \, dx\\ &=-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}+\frac{\int \frac{\sqrt{c+d x} \left (-\frac{3}{4} \left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right )-\frac{3}{2} b d (b c-5 a d) x\right )}{x^5 \sqrt{a+b x}} \, dx}{30 a}\\ &=\frac{\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{160 a^2 x^4}-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}+\frac{\int \frac{\frac{3}{8} \left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right )+\frac{3}{4} b d \left (9 b^2 c^2-26 a b c d+25 a^2 d^2\right ) x}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{120 a^2}\\ &=\frac{\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{160 a^2 x^4}-\frac{\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c x^3}-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\int \frac{\frac{3}{16} \left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right )+\frac{3}{4} b d \left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{360 a^3 c}\\ &=\frac{\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{160 a^2 x^4}-\frac{\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c x^3}+\frac{\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{3840 a^4 c^2 x^2}-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}+\frac{\int \frac{\frac{3}{32} \left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right )+\frac{3}{16} b d \left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{720 a^4 c^2}\\ &=\frac{\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{160 a^2 x^4}-\frac{\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c x^3}+\frac{\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{3840 a^4 c^2 x^2}-\frac{\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt{a+b x} \sqrt{c+d x}}{7680 a^5 c^3 x}-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\int \frac{45 (b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )}{64 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{720 a^5 c^3}\\ &=\frac{\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{160 a^2 x^4}-\frac{\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c x^3}+\frac{\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{3840 a^4 c^2 x^2}-\frac{\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt{a+b x} \sqrt{c+d x}}{7680 a^5 c^3 x}-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{1024 a^5 c^3}\\ &=\frac{\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{160 a^2 x^4}-\frac{\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c x^3}+\frac{\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{3840 a^4 c^2 x^2}-\frac{\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt{a+b x} \sqrt{c+d x}}{7680 a^5 c^3 x}-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 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 )}{512 a^5 c^3}\\ &=\frac{\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{160 a^2 x^4}-\frac{\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c x^3}+\frac{\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{3840 a^4 c^2 x^2}-\frac{\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt{a+b x} \sqrt{c+d x}}{7680 a^5 c^3 x}-\frac{(b c+5 a d) \sqrt{a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}+\frac{(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{11/2} c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.416215, size = 267, normalized size = 0.61 \[ \frac{\frac{\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \left (\frac{x (b c-a d) \left (\frac{5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}-8 \sqrt{a+b x} (c+d x)^{5/2}\right )}{a}-48 \sqrt{a+b x} (c+d x)^{7/2}\right )}{64 c x^4}+\frac{2 (a+b x)^{3/2} (c+d x)^{7/2} (5 a d+9 b c)}{x^5}-\frac{20 a c (a+b x)^{3/2} (c+d x)^{7/2}}{x^6}}{120 a^2 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]

[Out]

((-20*a*c*(a + b*x)^(3/2)*(c + d*x)^(7/2))/x^6 + (2*(9*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/x^5 + ((2
1*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*(-48*Sqrt[a + b*x]*(c + d*x)^(7/2) + ((b*c - a*d)*x*(-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]*Sqrt[c + d*x])]))/(a^(5/2)*Sqrt[c])))/a))/(64*c*x^4))
/(120*a^2*c^2)

________________________________________________________________________________________

Maple [B]  time = 0.028, size = 1271, normalized size = 2.9 \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)*(b*x+a)^(1/2)/x^7,x)

[Out]

1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^5/c^3*(-630*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*b^5*c^5+100*
(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^5*c*d^4+420*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*
a*b^4*c^5-80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^5*c^2*d^3-256*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*x*a^4*b*c^5-90*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^5*b*c*
d^5-75*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^4*b^2*c^2*d^4-300*ln((a*d
*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^3*b^3*c^3*d^3+1125*ln((a*d*x+b*c*x+2*(a
*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^2*b^4*c^4*d^2-1050*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a*b^5*c^5*d-336*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2
*b^3*c^5+288*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b^2*c^5-4320*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*x^2*a^5*c^3*d^2-150*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^5*d^5-6400*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*c^4*d-2560*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*c^5*(a*c)^(1/2)+75*ln((a*d*x+b
*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^6*d^6+315*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*b^6*c^6+130*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^4*b*c
*d^4+180*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^3*b^2*c^2*d^3-1676*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*x^5*a^2*b^3*c^3*d^2+1890*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a*b^4*c^4*d-80*(a*c)^(1/2
)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^4*b*c^2*d^3+1048*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b
^2*c^3*d^2-1232*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b^3*c^4*d-816*(a*c)^(1/2)*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*x^3*a^4*b*c^3*d^2+976*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b^2*c^4*d-832*(a*c)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^4*b*c^4*d)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^6/(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)*(b*x+a)^(1/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 140.092, size = 2014, normalized size = 4.62 \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)*(b*x+a)^(1/2)/x^7,x, algorithm="fricas")

[Out]

[1/30720*(15*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^
5*b*c*d^5 + 5*a^6*d^6)*sqrt(a*c)*x^6*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*(1280*a^6*c^6 + (315*a*b^5*c
^6 - 945*a^2*b^4*c^5*d + 838*a^3*b^3*c^4*d^2 - 90*a^4*b^2*c^3*d^3 - 65*a^5*b*c^2*d^4 + 75*a^6*c*d^5)*x^5 - 2*(
105*a^2*b^4*c^6 - 308*a^3*b^3*c^5*d + 262*a^4*b^2*c^4*d^2 - 20*a^5*b*c^3*d^3 + 25*a^6*c^2*d^4)*x^4 + 8*(21*a^3
*b^3*c^6 - 61*a^4*b^2*c^5*d + 51*a^5*b*c^4*d^2 + 5*a^6*c^3*d^3)*x^3 - 16*(9*a^4*b^2*c^6 - 26*a^5*b*c^5*d - 135
*a^6*c^4*d^2)*x^2 + 128*(a^5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*c^4*x^6), -1/15360*(15
*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 +
5*a^6*d^6)*sqrt(-a*c)*x^6*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*(1280*a^6*c^6 + (315*a*b^5*c^6 - 945*a^2*b^4*c^5*d + 838*a^3*b^3*c^
4*d^2 - 90*a^4*b^2*c^3*d^3 - 65*a^5*b*c^2*d^4 + 75*a^6*c*d^5)*x^5 - 2*(105*a^2*b^4*c^6 - 308*a^3*b^3*c^5*d + 2
62*a^4*b^2*c^4*d^2 - 20*a^5*b*c^3*d^3 + 25*a^6*c^2*d^4)*x^4 + 8*(21*a^3*b^3*c^6 - 61*a^4*b^2*c^5*d + 51*a^5*b*
c^4*d^2 + 5*a^6*c^3*d^3)*x^3 - 16*(9*a^4*b^2*c^6 - 26*a^5*b*c^5*d - 135*a^6*c^4*d^2)*x^2 + 128*(a^5*b*c^6 + 25
*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*c^4*x^6)]

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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)*(b*x+a)**(1/2)/x**7,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)*(b*x+a)^(1/2)/x^7,x, algorithm="giac")

[Out]

Exception raised: TypeError