3.659 \(\int \frac{(a+b x)^{5/2} (c+d x)^{3/2}}{x} \, dx\)

Optimal. Leaf size=299 \[ \frac{\left (90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{5/2}}-2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (64 a^2 b c d^2+(b c-5 a d) (b c-a d) (a d+3 b c)\right )}{64 b d^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-5 a d) (a d+3 b c)}{32 d^2}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 d} \]

[Out]

((64*a^2*b*c*d^2 + (b*c - 5*a*d)*(b*c - a*d)*(3*b*c + a*d))*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d^2) - ((b*c -
5*a*d)*(3*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*d^2) + ((3*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2
))/(24*d) + ((a + b*x)^(5/2)*(c + d*x)^(3/2))/4 - 2*a^(5/2)*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])] + ((3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.378458, antiderivative size = 294, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \[ \frac{1}{64} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{5 a^3 d}{b}+73 a^2 c-\frac{17 a b c^2}{d}+\frac{3 b^2 c^3}{d^2}\right )+\frac{\left (90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{5/2}}-2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-5 a d) (a d+3 b c)}{32 d^2}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x,x]

[Out]

((73*a^2*c + (3*b^2*c^3)/d^2 - (17*a*b*c^2)/d + (5*a^3*d)/b)*Sqrt[a + b*x]*Sqrt[c + d*x])/64 - ((b*c - 5*a*d)*
(3*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*d^2) + ((3*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*
d) + ((a + b*x)^(5/2)*(c + d*x)^(3/2))/4 - 2*a^(5/2)*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c +
d*x])] + ((3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqr
t[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(5/2))

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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} (c+d x)^{3/2}}{x} \, dx &=\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac{1}{4} \int \frac{(a+b x)^{3/2} \sqrt{c+d x} \left (-4 a c+\frac{1}{2} (-3 b c-5 a d) x\right )}{x} \, dx\\ &=\frac{(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac{\int \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-12 a^2 c d+\frac{3}{4} (b c-5 a d) (3 b c+a d) x\right )}{x} \, dx}{12 d}\\ &=-\frac{(b c-5 a d) (3 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 d^2}+\frac{(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac{\int \frac{\sqrt{c+d x} \left (-24 a^3 c d^2-\frac{3}{8} \left (64 a^2 b c d^2+(b c-5 a d) (b c-a d) (3 b c+a d)\right ) x\right )}{x \sqrt{a+b x}} \, dx}{24 d^2}\\ &=\frac{1}{64} \left (73 a^2 c+\frac{3 b^2 c^3}{d^2}-\frac{17 a b c^2}{d}+\frac{5 a^3 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}-\frac{(b c-5 a d) (3 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 d^2}+\frac{(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-\frac{\int \frac{-24 a^3 b c^2 d^2-\frac{3}{16} \left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 b d^2}\\ &=\frac{1}{64} \left (73 a^2 c+\frac{3 b^2 c^3}{d^2}-\frac{17 a b c^2}{d}+\frac{5 a^3 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}-\frac{(b c-5 a d) (3 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 d^2}+\frac{(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\left (a^3 c^2\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx+\frac{\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b d^2}\\ &=\frac{1}{64} \left (73 a^2 c+\frac{3 b^2 c^3}{d^2}-\frac{17 a b c^2}{d}+\frac{5 a^3 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}-\frac{(b c-5 a d) (3 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 d^2}+\frac{(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\left (2 a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )+\frac{\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{64 b^2 d^2}\\ &=\frac{1}{64} \left (73 a^2 c+\frac{3 b^2 c^3}{d^2}-\frac{17 a b c^2}{d}+\frac{5 a^3 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}-\frac{(b c-5 a d) (3 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 d^2}+\frac{(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 b^2 d^2}\\ &=\frac{1}{64} \left (73 a^2 c+\frac{3 b^2 c^3}{d^2}-\frac{17 a b c^2}{d}+\frac{5 a^3 d}{b}\right ) \sqrt{a+b x} \sqrt{c+d x}-\frac{(b c-5 a d) (3 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{32 d^2}+\frac{(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 d}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}-2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.07763, size = 300, normalized size = 1. \[ \frac{\sqrt{d} \left (\sqrt{a+b x} (c+d x) \left (a^2 b d^2 (337 c+118 d x)+15 a^3 d^3+a b^2 d \left (57 c^2+244 c d x+136 d^2 x^2\right )+b^3 \left (6 c^2 d x-9 c^3+72 c d^2 x^2+48 d^3 x^3\right )\right )-384 a^{5/2} b c^{3/2} d^2 \sqrt{c+d x} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )+\frac{3 \sqrt{b c-a d} \left (90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c^3 d+3 b^4 c^4\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b}}{192 b d^{5/2} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x,x]

[Out]

((3*Sqrt[b*c - a*d]*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*Sqrt[(b*(c
+ d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/b + Sqrt[d]*(Sqrt[a + b*x]*(c + d*x)*(1
5*a^3*d^3 + a^2*b*d^2*(337*c + 118*d*x) + a*b^2*d*(57*c^2 + 244*c*d*x + 136*d^2*x^2) + b^3*(-9*c^3 + 6*c^2*d*x
+ 72*c*d^2*x^2 + 48*d^3*x^3)) - 384*a^(5/2)*b*c^(3/2)*d^2*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt
[a]*Sqrt[c + d*x])]))/(192*b*d^(5/2)*Sqrt[c + d*x])

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 828, normalized size = 2.8 \begin{align*} -{\frac{1}{384\,b{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-272\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-144\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{4}{d}^{4}-180\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{3}bc{d}^{3}-270\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+60\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}a{b}^{3}{c}^{3}d-9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{b}^{4}{c}^{4}+384\,\sqrt{bd}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){a}^{3}b{c}^{2}{d}^{2}-236\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{d}^{3}-488\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{2}c{d}^{2}-12\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{3}{c}^{2}d-30\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{d}^{3}-674\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}bc{d}^{2}-114\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{2}{c}^{2}d+18\,\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{3}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x,x)

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*x^3*b^3*d^3*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-27
2*x^2*a*b^2*d^3*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-144*x^2*b^3*c*d^2*(a*c)^(1/2)*(b*d*x^2
+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+15*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*(a*c)^(1/2)*a^4*d^4-180*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b
*d)^(1/2))*(a*c)^(1/2)*a^3*b*c*d^3-270*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*(a*c)^(1/2)*a^2*b^2*c^2*d^2+60*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+
b*c)/(b*d)^(1/2))*(a*c)^(1/2)*a*b^3*c^3*d-9*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+
b*c)/(b*d)^(1/2))*(a*c)^(1/2)*b^4*c^4+384*(b*d)^(1/2)*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)*a^3*b*c^2*d^2-236*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*d^3-488*(b*d
)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^2*c*d^2-12*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*x*b^3*c^2*d-30*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*d^3-674*(b*d)^(1/2)*
(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b*c*d^2-114*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*a*b^2*c^2*d+18*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^3*c^3)/b/d^2/(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)/(b*d)^(1/2)/(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((b*x+a)^(5/2)*(d*x+c)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 112.743, size = 3359, normalized size = 11.23 \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)*(d*x+c)^(3/2)/x,x, algorithm="fricas")

[Out]

[1/768*(384*sqrt(a*c)*a^2*b^2*c*d^3*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) - 3*(3*b^4*c^4 - 20*a*b^3*c^3*d +
90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2
- 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(48*b^4*d^4*x^3
- 9*b^4*c^3*d + 57*a*b^3*c^2*d^2 + 337*a^2*b^2*c*d^3 + 15*a^3*b*d^4 + 8*(9*b^4*c*d^3 + 17*a*b^3*d^4)*x^2 + 2*
(3*b^4*c^2*d^2 + 122*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^3), 1/384*(192*sqrt(
a*c)*a^2*b^2*c*d^3*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) - 3*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d
^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x
+ c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(48*b^4*d^4*x^3 - 9*b^4*c^3*d + 57*a*b^3*c^2*d^2 +
337*a^2*b^2*c*d^3 + 15*a^3*b*d^4 + 8*(9*b^4*c*d^3 + 17*a*b^3*d^4)*x^2 + 2*(3*b^4*c^2*d^2 + 122*a*b^3*c*d^3 + 5
9*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^3), 1/768*(768*sqrt(-a*c)*a^2*b^2*c*d^3*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)) - 3
*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt(b*d)*log(8*b^2*d^2*x^2 +
b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a
*b*d^2)*x) + 4*(48*b^4*d^4*x^3 - 9*b^4*c^3*d + 57*a*b^3*c^2*d^2 + 337*a^2*b^2*c*d^3 + 15*a^3*b*d^4 + 8*(9*b^4*
c*d^3 + 17*a*b^3*d^4)*x^2 + 2*(3*b^4*c^2*d^2 + 122*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c
))/(b^2*d^3), 1/384*(384*sqrt(-a*c)*a^2*b^2*c*d^3*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)) - 3*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^
2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(
d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(48*b^4*d^4*x^3 - 9*b^4*c^3*d + 57*a*b^3*c^2*d^2
+ 337*a^2*b^2*c*d^3 + 15*a^3*b*d^4 + 8*(9*b^4*c*d^3 + 17*a*b^3*d^4)*x^2 + 2*(3*b^4*c^2*d^2 + 122*a*b^3*c*d^3
+ 59*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^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)*(d*x+c)**(3/2)/x,x)

[Out]

Timed out

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Giac [A]  time = 1.66491, size = 574, normalized size = 1.92 \begin{align*} -\frac{2 \, \sqrt{b d} a^{3} c^{2}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{192} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} d{\left | b \right |}}{b^{3}} + \frac{9 \, b^{6} c d^{6}{\left | b \right |} - a b^{5} d^{7}{\left | b \right |}}{b^{8} d^{6}}\right )} + \frac{3 \, b^{7} c^{2} d^{5}{\left | b \right |} + 50 \, a b^{6} c d^{6}{\left | b \right |} - 5 \, a^{2} b^{5} d^{7}{\left | b \right |}}{b^{8} d^{6}}\right )} - \frac{3 \,{\left (3 \, b^{8} c^{3} d^{4}{\left | b \right |} - 17 \, a b^{7} c^{2} d^{5}{\left | b \right |} - 55 \, a^{2} b^{6} c d^{6}{\left | b \right |} + 5 \, a^{3} b^{5} d^{7}{\left | b \right |}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} - \frac{{\left (3 \, \sqrt{b d} b^{4} c^{4}{\left | b \right |} - 20 \, \sqrt{b d} a b^{3} c^{3} d{\left | b \right |} + 90 \, \sqrt{b d} a^{2} b^{2} c^{2} d^{2}{\left | b \right |} + 60 \, \sqrt{b d} a^{3} b c d^{3}{\left | b \right |} - 5 \, \sqrt{b d} a^{4} d^{4}{\left | b \right |}\right )} \log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x,x, algorithm="giac")

[Out]

-2*sqrt(b*d)*a^3*c^2*abs(b)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b) + 1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*
(4*(b*x + a)*(6*(b*x + a)*d*abs(b)/b^3 + (9*b^6*c*d^6*abs(b) - a*b^5*d^7*abs(b))/(b^8*d^6)) + (3*b^7*c^2*d^5*a
bs(b) + 50*a*b^6*c*d^6*abs(b) - 5*a^2*b^5*d^7*abs(b))/(b^8*d^6)) - 3*(3*b^8*c^3*d^4*abs(b) - 17*a*b^7*c^2*d^5*
abs(b) - 55*a^2*b^6*c*d^6*abs(b) + 5*a^3*b^5*d^7*abs(b))/(b^8*d^6))*sqrt(b*x + a) - 1/128*(3*sqrt(b*d)*b^4*c^4
*abs(b) - 20*sqrt(b*d)*a*b^3*c^3*d*abs(b) + 90*sqrt(b*d)*a^2*b^2*c^2*d^2*abs(b) + 60*sqrt(b*d)*a^3*b*c*d^3*abs
(b) - 5*sqrt(b*d)*a^4*d^4*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^3*
d^3)