3.565 \(\int \frac{\sqrt{a+b x} (c+d x)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=340 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^2 b c d^2-15 a^3 d^3-61 a b^2 c^2 d+35 b^3 c^3\right )}{960 a^3 c^2 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-190 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^3 x}-\frac{\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{7 b^2 c}{a}-\frac{3 a d^2}{c}-12 b d\right )}{240 a x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{40 a x^4} \]

[Out]

-((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a*x^4) + (((7*b^2*c)/a - 12*b*d - (3*a*d^2)/c)*Sqrt[a + b*x]*
Sqrt[c + d*x])/(240*a*x^3) - ((35*b^3*c^3 - 61*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(960*a^3*c^2*x^2) + ((105*b^4*c^4 - 190*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 30*a^3*b*c*d^3 - 45*a^4*d^4
)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^4*c^3*x) - (Sqrt[a + b*x]*(c + d*x)^(3/2))/(5*x^5) - ((b*c - a*d)^3*(7*
b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(7/2
))

________________________________________________________________________________________

Rubi [A]  time = 0.326844, antiderivative size = 340, 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 = {97, 149, 151, 12, 93, 208} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^2 b c d^2-15 a^3 d^3-61 a b^2 c^2 d+35 b^3 c^3\right )}{960 a^3 c^2 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-190 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^3 x}-\frac{\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{7 b^2 c}{a}-\frac{3 a d^2}{c}-12 b d\right )}{240 a x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{40 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^6,x]

[Out]

-((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a*x^4) + (((7*b^2*c)/a - 12*b*d - (3*a*d^2)/c)*Sqrt[a + b*x]*
Sqrt[c + d*x])/(240*a*x^3) - ((35*b^3*c^3 - 61*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(960*a^3*c^2*x^2) + ((105*b^4*c^4 - 190*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 30*a^3*b*c*d^3 - 45*a^4*d^4
)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^4*c^3*x) - (Sqrt[a + b*x]*(c + d*x)^(3/2))/(5*x^5) - ((b*c - a*d)^3*(7*
b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*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)^{3/2}}{x^6} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}+\frac{1}{5} \int \frac{\sqrt{c+d x} \left (\frac{1}{2} (b c+3 a d)+2 b d x\right )}{x^5 \sqrt{a+b x}} \, dx\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{40 a x^4}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}+\frac{\int \frac{\frac{1}{4} \left (-7 b^2 c^2+12 a b c d+3 a^2 d^2\right )-\frac{1}{2} b d (3 b c-7 a d) x}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{20 a}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{40 a x^4}+\frac{\left (\frac{7 b^2 c}{a}-12 b d-\frac{3 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 a x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}-\frac{\int \frac{\frac{1}{8} \left (-35 b^3 c^3+61 a b^2 c^2 d-9 a^2 b c d^2+15 a^3 d^3\right )-\frac{1}{2} b d \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{60 a^2 c}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{40 a x^4}+\frac{\left (\frac{7 b^2 c}{a}-12 b d-\frac{3 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 a x^3}-\frac{\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c^2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}+\frac{\int \frac{\frac{1}{16} \left (-105 b^4 c^4+190 a b^3 c^3 d-36 a^2 b^2 c^2 d^2-30 a^3 b c d^3+45 a^4 d^4\right )-\frac{1}{8} b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 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^3 c^2}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{40 a x^4}+\frac{\left (\frac{7 b^2 c}{a}-12 b d-\frac{3 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 a x^3}-\frac{\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c^2 x^2}+\frac{\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^4 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}-\frac{\int -\frac{15 (b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )}{32 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{120 a^4 c^3}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{40 a x^4}+\frac{\left (\frac{7 b^2 c}{a}-12 b d-\frac{3 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 a x^3}-\frac{\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c^2 x^2}+\frac{\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^4 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}+\frac{\left ((b c-a d)^3 \left (7 b^2 c^2+6 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^4 c^3}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{40 a x^4}+\frac{\left (\frac{7 b^2 c}{a}-12 b d-\frac{3 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 a x^3}-\frac{\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c^2 x^2}+\frac{\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^4 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}+\frac{\left ((b c-a d)^3 \left (7 b^2 c^2+6 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^4 c^3}\\ &=-\frac{(b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{40 a x^4}+\frac{\left (\frac{7 b^2 c}{a}-12 b d-\frac{3 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 a x^3}-\frac{\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^3 c^2 x^2}+\frac{\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^4 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}-\frac{(b c-a d)^3 \left (7 b^2 c^2+6 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^{9/2} c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.293289, size = 232, normalized size = 0.68 \[ \frac{-\frac{5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \left (\frac{x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}+8 \sqrt{a+b x} (c+d x)^{5/2}\right )}{24 c x^3}+\frac{2 (a+b x)^{3/2} (c+d x)^{5/2} (5 a d+7 b c)}{x^4}-\frac{16 a c (a+b x)^{3/2} (c+d x)^{5/2}}{x^5}}{80 a^2 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^6,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.018, size = 967, normalized size = 2.8 \begin{align*}{\frac{1}{3840\,{a}^{4}{c}^{3}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}-90\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}+225\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d-105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-90\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}+72\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-380\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d+210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-36\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}+244\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d-140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-48\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}-192\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d+112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}-1056\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d-96\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}-768\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^6,x)

[Out]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3*(45*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^5*a^5*d^5-45*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^5*a^4*b*c*d^
4-30*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^5*a^3*b^2*c^2*d^3-90*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^5*a^2*b^3*c^3*d^2+225*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^5*a*b^4*c^4*d-105*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^5*b^5*c^5-90*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^4*d^4+60*(a*c)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b*c*d^3+72*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b
^2*c^2*d^2-380*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^3*c^3*d+210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*x^4*b^4*c^4+60*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^4*c*d^3-36*(a*c)^(1/2)*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b*c^2*d^2+244*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^2*c^3*d-140
*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^3*c^4-48*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*
a^4*c^2*d^2-192*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b*c^3*d+112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^2*a^2*b^2*c^4-1056*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c^3*d-96*(a*c)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*b*c^4-768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^4*(a*c)^(1/2))/(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)/x^5/(a*c)^(1/2)

________________________________________________________________________________________

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)^(3/2)*(b*x+a)^(1/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 80.9374, size = 1596, normalized size = 4.69 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, 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 (105 \, a b^{4} c^{5} - 190 \, a^{2} b^{3} c^{4} d + 36 \, a^{3} b^{2} c^{3} d^{2} + 30 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \,{\left (35 \, a^{2} b^{3} c^{5} - 61 \, a^{3} b^{2} c^{4} d + 9 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{3} b^{2} c^{5} - 12 \, a^{4} b c^{4} d - 3 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \,{\left (a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, a^{5} c^{4} x^{5}}, \frac{15 \,{\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, 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 (105 \, a b^{4} c^{5} - 190 \, a^{2} b^{3} c^{4} d + 36 \, a^{3} b^{2} c^{3} d^{2} + 30 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \,{\left (35 \, a^{2} b^{3} c^{5} - 61 \, a^{3} b^{2} c^{4} d + 9 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{3} b^{2} c^{5} - 12 \, a^{4} b c^{4} d - 3 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \,{\left (a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, a^{5} c^{4} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[-1/7680*(15*(7*b^5*c^5 - 15*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 + 3*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 - (105*a*b^4*c^5 - 190*a^2*b^3*c^4*d +
36*a^3*b^2*c^3*d^2 + 30*a^4*b*c^2*d^3 - 45*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 61*a^3*b^2*c^4*d + 9*a^4*b*c^
3*d^2 - 15*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 12*a^4*b*c^4*d - 3*a^5*c^3*d^2)*x^2 + 48*(a^4*b*c^5 + 11*a^5*
c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^4*x^5), 1/3840*(15*(7*b^5*c^5 - 15*a*b^4*c^4*d + 6*a^2*b^3*c^3*d
^2 + 2*a^3*b^2*c^2*d^3 + 3*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*(384*a^5*c^5 - (105*a*b^4*
c^5 - 190*a^2*b^3*c^4*d + 36*a^3*b^2*c^3*d^2 + 30*a^4*b*c^2*d^3 - 45*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 61*a
^3*b^2*c^4*d + 9*a^4*b*c^3*d^2 - 15*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 12*a^4*b*c^4*d - 3*a^5*c^3*d^2)*x^2
+ 48*(a^4*b*c^5 + 11*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^4*x^5)]

________________________________________________________________________________________

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)**(3/2)*(b*x+a)**(1/2)/x**6,x)

[Out]

Timed out

________________________________________________________________________________________

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)^(3/2)*(b*x+a)^(1/2)/x^6,x, algorithm="giac")

[Out]

Exception raised: TypeError