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

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

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Rubi [A]  time = 0.38, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 1.10, size = 300, normalized size = 1.00 \begin {gather*} \frac {\sqrt {d} \left (\sqrt {a+b x} (c+d x) \left (15 a^3 d^3+a^2 b d^2 (337 c+118 d x)+a b^2 d \left (57 c^2+244 c d x+136 d^2 x^2\right )+b^3 \left (-9 c^3+6 c^2 d x+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 (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 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}} \end {gather*}

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])

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IntegrateAlgebraic [B]  time = 0.72, size = 754, normalized size = 2.52 \begin {gather*} -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 (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 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}}+\frac {\frac {15 a^4 b^3 d^4 \sqrt {a+b x}}{\sqrt {c+d x}}-\frac {55 a^4 b^2 d^5 (a+b x)^{3/2}}{(c+d x)^{3/2}}+\frac {15 a^4 d^7 (a+b x)^{7/2}}{(c+d x)^{7/2}}+\frac {73 a^4 b d^6 (a+b x)^{5/2}}{(c+d x)^{5/2}}-\frac {180 a^3 b^4 c d^3 \sqrt {a+b x}}{\sqrt {c+d x}}+\frac {660 a^3 b^3 c d^4 (a+b x)^{3/2}}{(c+d x)^{3/2}}-\frac {876 a^3 b^2 c d^5 (a+b x)^{5/2}}{(c+d x)^{5/2}}+\frac {204 a^3 b c d^6 (a+b x)^{7/2}}{(c+d x)^{7/2}}+\frac {114 a^2 b^5 c^2 d^2 \sqrt {a+b x}}{\sqrt {c+d x}}-\frac {546 a^2 b^4 c^2 d^3 (a+b x)^{3/2}}{(c+d x)^{3/2}}+\frac {990 a^2 b^3 c^2 d^4 (a+b x)^{5/2}}{(c+d x)^{5/2}}-\frac {270 a^2 b^2 c^2 d^5 (a+b x)^{7/2}}{(c+d x)^{7/2}}-\frac {9 b^7 c^4 \sqrt {a+b x}}{\sqrt {c+d x}}+\frac {33 b^6 c^4 d (a+b x)^{3/2}}{(c+d x)^{3/2}}+\frac {60 a b^6 c^3 d \sqrt {a+b x}}{\sqrt {c+d x}}+\frac {33 b^5 c^4 d^2 (a+b x)^{5/2}}{(c+d x)^{5/2}}-\frac {92 a b^5 c^3 d^2 (a+b x)^{3/2}}{(c+d x)^{3/2}}-\frac {9 b^4 c^4 d^3 (a+b x)^{7/2}}{(c+d x)^{7/2}}-\frac {220 a b^4 c^3 d^3 (a+b x)^{5/2}}{(c+d x)^{5/2}}+\frac {60 a b^3 c^3 d^4 (a+b x)^{7/2}}{(c+d x)^{7/2}}}{192 b d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-9*b^4*c^4*d^3*(a + b*x)^(7/2))/(c + d*x)^(7/2) + (60*a*b^3*c^3*d^4*(a + b*x)^(7/2))/(c + d*x)^(7/2) - (270*
a^2*b^2*c^2*d^5*(a + b*x)^(7/2))/(c + d*x)^(7/2) + (204*a^3*b*c*d^6*(a + b*x)^(7/2))/(c + d*x)^(7/2) + (15*a^4
*d^7*(a + b*x)^(7/2))/(c + d*x)^(7/2) + (33*b^5*c^4*d^2*(a + b*x)^(5/2))/(c + d*x)^(5/2) - (220*a*b^4*c^3*d^3*
(a + b*x)^(5/2))/(c + d*x)^(5/2) + (990*a^2*b^3*c^2*d^4*(a + b*x)^(5/2))/(c + d*x)^(5/2) - (876*a^3*b^2*c*d^5*
(a + b*x)^(5/2))/(c + d*x)^(5/2) + (73*a^4*b*d^6*(a + b*x)^(5/2))/(c + d*x)^(5/2) + (33*b^6*c^4*d*(a + b*x)^(3
/2))/(c + d*x)^(3/2) - (92*a*b^5*c^3*d^2*(a + b*x)^(3/2))/(c + d*x)^(3/2) - (546*a^2*b^4*c^2*d^3*(a + b*x)^(3/
2))/(c + d*x)^(3/2) + (660*a^3*b^3*c*d^4*(a + b*x)^(3/2))/(c + d*x)^(3/2) - (55*a^4*b^2*d^5*(a + b*x)^(3/2))/(
c + d*x)^(3/2) - (9*b^7*c^4*Sqrt[a + b*x])/Sqrt[c + d*x] + (60*a*b^6*c^3*d*Sqrt[a + b*x])/Sqrt[c + d*x] + (114
*a^2*b^5*c^2*d^2*Sqrt[a + b*x])/Sqrt[c + d*x] - (180*a^3*b^4*c*d^3*Sqrt[a + b*x])/Sqrt[c + d*x] + (15*a^4*b^3*
d^4*Sqrt[a + b*x])/Sqrt[c + d*x])/(192*b*d^2*(b - (d*(a + b*x))/(c + d*x))^4) - 2*a^(5/2)*c^(3/2)*ArcTanh[(Sqr
t[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]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(5/2))

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fricas [A]  time = 44.02, size = 1481, normalized size = 4.95

result too large to display

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 1.08

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maple [B]  time = 0.02, size = 828, normalized size = 2.77 \begin {gather*} -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \sqrt {a c}\, a^{4} d^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+384 \sqrt {b d}\, a^{3} b \,c^{2} d^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-180 \sqrt {a c}\, a^{3} b c \,d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-270 \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+60 \sqrt {a c}\, a \,b^{3} c^{3} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-9 \sqrt {a c}\, b^{4} c^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} d^{3} x^{3}-272 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} d^{3} x^{2}-144 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c \,d^{2} x^{2}-236 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b \,d^{3} x -488 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{2} c \,d^{2} x -12 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{3} c^{2} d x -30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} d^{3}-674 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b c \,d^{2}-114 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{2} c^{2} d +18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{3} c^{3}\right )}{384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b \,d^{2}} \end {gather*}

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*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*b^3*d^3*x^3-27
2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^2*d^3*x^2-144*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b
*d)^(1/2)*(a*c)^(1/2)*b^3*c*d^2*x^2+15*(a*c)^(1/2)*a^4*d^4*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-180*(a*c)^(1/2)*a^3*b*c*d^3*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-270*(a*c)^(1/2)*a^2*b^2*c^2*d^2*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+60*(a*c)^(1/2)*a*b^3*c^3*d*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-9*(a*c)^(1/2)*b^4*c^4*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+384*(b*d)^(1/2)*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2))/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)*a^2*b*d^3*x-488*(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*d^2*x-12*(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^2*d*x-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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}

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]

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

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