3.6.97 \(\int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx\) [597]

Optimal. Leaf size=234 \[ -\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}+\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}} \]

[Out]

1/4*(-35*a^2*d^2+10*a*b*c*d+b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(3/2)/c^(9/2)-1/12
*d*(-35*a*d+3*b*c)*(b*x+a)^(1/2)/a/c^3/(d*x+c)^(3/2)-1/2*(b*x+a)^(1/2)/c/x^2/(d*x+c)^(3/2)-1/4*(-7*a*d+b*c)*(b
*x+a)^(1/2)/a/c^2/x/(d*x+c)^(3/2)-1/12*d*(105*a^2*d^2-100*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)/a/c^4/(-a*d+b*c)/(d
*x+c)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {101, 156, 157, 12, 95, 214} \begin {gather*} -\frac {d \sqrt {a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{12 a c^4 \sqrt {c+d x} (b c-a d)}+\frac {\left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}}-\frac {d \sqrt {a+b x} (3 b c-35 a d)}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(d*(3*b*c - 35*a*d)*Sqrt[a + b*x])/(a*c^3*(c + d*x)^(3/2)) - Sqrt[a + b*x]/(2*c*x^2*(c + d*x)^(3/2)) - (
(b*c - 7*a*d)*Sqrt[a + b*x])/(4*a*c^2*x*(c + d*x)^(3/2)) - (d*(3*b^2*c^2 - 100*a*b*c*d + 105*a^2*d^2)*Sqrt[a +
b*x])/(12*a*c^4*(b*c - a*d)*Sqrt[c + d*x]) + ((b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b
*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(9/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

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 + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

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] && ILtQ[m, -1]

Rule 157

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 214

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{5/2}} \, dx &=-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}+\frac {\int \frac {\frac {1}{2} (b c-7 a d)-3 b d x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 c}\\ &=-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{4} \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )+b d (b c-7 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 a c^2}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}+\frac {\int \frac {-\frac {3}{8} (b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )-\frac {1}{4} b d (3 b c-35 a d) (b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a c^3 (b c-a d)}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {3 (b c-a d)^2 \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c^4 (b c-a d)^2}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a c^4}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a c^4}\\ &=-\frac {d (3 b c-35 a d) \sqrt {a+b x}}{12 a c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(b c-7 a d) \sqrt {a+b x}}{4 a c^2 x (c+d x)^{3/2}}-\frac {d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt {a+b x}}{12 a c^4 (b c-a d) \sqrt {c+d x}}+\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 195, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {a+b x} \left (3 b^2 c^2 x (c+d x)^2+2 a b c \left (3 c^3-12 c^2 d x-69 c d^2 x^2-50 d^3 x^3\right )+a^2 d \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )\right )}{12 a c^4 (b c-a d) x^2 (c+d x)^{3/2}}+\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(Sqrt[a + b*x]*(3*b^2*c^2*x*(c + d*x)^2 + 2*a*b*c*(3*c^3 - 12*c^2*d*x - 69*c*d^2*x^2 - 50*d^3*x^3) + a^2
*d*(-6*c^3 + 21*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3)))/(a*c^4*(b*c - a*d)*x^2*(c + d*x)^(3/2)) + ((b^2*c^2 +
10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(196)=392\).
time = 0.07, size = 988, normalized size = 4.22

method	result	size
default	\(-\frac {\left (105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{5} x^{4}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{4} x^{4}+27 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d^{3} x^{4}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} d^{2} x^{4}+210 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c \,d^{4} x^{3}-270 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d^{3} x^{3}+54 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} d^{2} x^{3}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{4} d \,x^{3}+105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c^{2} d^{3} x^{2}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} d^{2} x^{2}+27 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{4} d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{5} x^{2}-210 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} d^{4} x^{3}+200 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a b c \,d^{3} x^{3}-6 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{2} c^{2} d^{2} x^{3}-280 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} c \,d^{3} x^{2}+276 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a b \,c^{2} d^{2} x^{2}-12 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{2} c^{3} d \,x^{2}-42 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} c^{2} d^{2} x +48 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a b \,c^{3} d x -6 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{2} c^{4} x +12 a^{2} c^{3} d \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-12 a b \,c^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right ) \sqrt {b x +a}}{24 c^{4} a \sqrt {a c}\, x^{2} \left (a d -b c \right ) \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \left (d x +c \right )^{\frac {3}{2}}}\)	\(988\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)/x^3/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*(105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*d^5*x^4-135*ln((a*d*x+b*c*x+2*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b*c*d^4*x^4+27*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a)
)^(1/2)+2*a*c)/x)*a*b^2*c^2*d^3*x^4+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^3*c^3*
d^2*x^4+210*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*c*d^4*x^3-270*ln((a*d*x+b*c*x+
2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b*c^2*d^3*x^3+54*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b
*x+a))^(1/2)+2*a*c)/x)*a*b^2*c^3*d^2*x^3+6*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^3
*c^4*d*x^3+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*c^2*d^3*x^2-135*ln((a*d*x+b
*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b*c^3*d^2*x^2+27*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+
c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b^2*c^4*d*x^2+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*
b^3*c^5*x^2-210*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*d^4*x^3+200*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a*b*c*
d^3*x^3-6*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b^2*c^2*d^2*x^3-280*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*c*d^
3*x^2+276*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a*b*c^2*d^2*x^2-12*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b^2*c^3*d
*x^2-42*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*c^2*d^2*x+48*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a*b*c^3*d*x-6
*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b^2*c^4*x+12*a^2*c^3*d*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-12*a*b*c^4*(a*
c)^(1/2)*((d*x+c)*(b*x+a))^(1/2))/c^4/a*(b*x+a)^(1/2)/(a*c)^(1/2)/x^2/(a*d-b*c)/((d*x+c)*(b*x+a))^(1/2)/(d*x+c
)^(3/2)

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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)^(1/2)/x^3/(d*x+c)^(5/2),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 detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (196) = 392\).
time = 3.40, size = 904, normalized size = 3.86 \begin {gather*} \left [-\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d + 9 \, a b^{2} c^{3} d^{2} - 45 \, a^{2} b c^{2} d^{3} + 35 \, a^{3} c d^{4}\right )} x^{3} + {\left (b^{3} c^{5} + 9 \, a b^{2} c^{4} d - 45 \, a^{2} b c^{3} d^{2} + 35 \, a^{3} c^{2} d^{3}\right )} x^{2}\right )} \sqrt {a c} \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 (6 \, a^{2} b c^{5} - 6 \, a^{3} c^{4} d + {\left (3 \, a b^{2} c^{3} d^{2} - 100 \, a^{2} b c^{2} d^{3} + 105 \, a^{3} c d^{4}\right )} x^{3} + 2 \, {\left (3 \, a b^{2} c^{4} d - 69 \, a^{2} b c^{3} d^{2} + 70 \, a^{3} c^{2} d^{3}\right )} x^{2} + 3 \, {\left (a b^{2} c^{5} - 8 \, a^{2} b c^{4} d + 7 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left ({\left (a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3}\right )} x^{4} + 2 \, {\left (a^{2} b c^{7} d - a^{3} c^{6} d^{2}\right )} x^{3} + {\left (a^{2} b c^{8} - a^{3} c^{7} d\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d + 9 \, a b^{2} c^{3} d^{2} - 45 \, a^{2} b c^{2} d^{3} + 35 \, a^{3} c d^{4}\right )} x^{3} + {\left (b^{3} c^{5} + 9 \, a b^{2} c^{4} d - 45 \, a^{2} b c^{3} d^{2} + 35 \, a^{3} c^{2} d^{3}\right )} x^{2}\right )} \sqrt {-a c} \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 (6 \, a^{2} b c^{5} - 6 \, a^{3} c^{4} d + {\left (3 \, a b^{2} c^{3} d^{2} - 100 \, a^{2} b c^{2} d^{3} + 105 \, a^{3} c d^{4}\right )} x^{3} + 2 \, {\left (3 \, a b^{2} c^{4} d - 69 \, a^{2} b c^{3} d^{2} + 70 \, a^{3} c^{2} d^{3}\right )} x^{2} + 3 \, {\left (a b^{2} c^{5} - 8 \, a^{2} b c^{4} d + 7 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left ({\left (a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3}\right )} x^{4} + 2 \, {\left (a^{2} b c^{7} d - a^{3} c^{6} d^{2}\right )} x^{3} + {\left (a^{2} b c^{8} - a^{3} c^{7} d\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/x^3/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(3*((b^3*c^3*d^2 + 9*a*b^2*c^2*d^3 - 45*a^2*b*c*d^4 + 35*a^3*d^5)*x^4 + 2*(b^3*c^4*d + 9*a*b^2*c^3*d^2
- 45*a^2*b*c^2*d^3 + 35*a^3*c*d^4)*x^3 + (b^3*c^5 + 9*a*b^2*c^4*d - 45*a^2*b*c^3*d^2 + 35*a^3*c^2*d^3)*x^2)*sq
rt(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*(6*a^2*b*c^5 - 6*a^3*c^4*d + (3*a*b^2*c^3*d^2 - 100*a^2*b
*c^2*d^3 + 105*a^3*c*d^4)*x^3 + 2*(3*a*b^2*c^4*d - 69*a^2*b*c^3*d^2 + 70*a^3*c^2*d^3)*x^2 + 3*(a*b^2*c^5 - 8*a
^2*b*c^4*d + 7*a^3*c^3*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^2*b*c^6*d^2 - a^3*c^5*d^3)*x^4 + 2*(a^2*b*c^7*
d - a^3*c^6*d^2)*x^3 + (a^2*b*c^8 - a^3*c^7*d)*x^2), -1/24*(3*((b^3*c^3*d^2 + 9*a*b^2*c^2*d^3 - 45*a^2*b*c*d^4
+ 35*a^3*d^5)*x^4 + 2*(b^3*c^4*d + 9*a*b^2*c^3*d^2 - 45*a^2*b*c^2*d^3 + 35*a^3*c*d^4)*x^3 + (b^3*c^5 + 9*a*b^
2*c^4*d - 45*a^2*b*c^3*d^2 + 35*a^3*c^2*d^3)*x^2)*sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqr
t(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*(6*a^2*b*c^5 - 6*a^3*c^4*d + (3*
a*b^2*c^3*d^2 - 100*a^2*b*c^2*d^3 + 105*a^3*c*d^4)*x^3 + 2*(3*a*b^2*c^4*d - 69*a^2*b*c^3*d^2 + 70*a^3*c^2*d^3)
*x^2 + 3*(a*b^2*c^5 - 8*a^2*b*c^4*d + 7*a^3*c^3*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^2*b*c^6*d^2 - a^3*c^5
*d^3)*x^4 + 2*(a^2*b*c^7*d - a^3*c^6*d^2)*x^3 + (a^2*b*c^8 - a^3*c^7*d)*x^2)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (196) = 392\).
time = 2.51, size = 1204, normalized size = 5.15 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (8 \, b^{4} c^{5} d^{4} {\left | b \right |} - 9 \, a b^{3} c^{4} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{9} d - a b^{2} c^{8} d^{2}} + \frac {9 \, {\left (b^{5} c^{6} d^{3} {\left | b \right |} - 2 \, a b^{4} c^{5} d^{4} {\left | b \right |} + a^{2} b^{3} c^{4} d^{5} {\left | b \right |}\right )}}{b^{3} c^{9} d - a b^{2} c^{8} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (\sqrt {b d} b^{4} c^{2} + 10 \, \sqrt {b d} a b^{3} c d - 35 \, \sqrt {b d} a^{2} b^{2} d^{2}\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 )}{4 \, \sqrt {-a b c d} a b c^{4} {\left | b \right |}} - \frac {\sqrt {b d} b^{10} c^{5} - 15 \, \sqrt {b d} a b^{9} c^{4} d + 50 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} - 70 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} + 45 \, \sqrt {b d} a^{4} b^{6} c d^{4} - 11 \, \sqrt {b d} a^{5} b^{5} d^{5} - 3 \, \sqrt {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} b^{8} c^{4} + 40 \, \sqrt {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} a b^{7} c^{3} d - 38 \, \sqrt {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} a^{2} b^{6} c^{2} d^{2} - 32 \, \sqrt {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} a^{3} b^{5} c d^{3} + 33 \, \sqrt {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} a^{4} b^{4} d^{4} + 3 \, \sqrt {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 )}^{4} b^{6} c^{3} - 31 \, \sqrt {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 )}^{4} a b^{5} c^{2} d - 19 \, \sqrt {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 )}^{4} a^{2} b^{4} c d^{2} - 33 \, \sqrt {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 )}^{4} a^{3} b^{3} d^{3} - \sqrt {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 )}^{6} b^{4} c^{2} + 6 \, \sqrt {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 )}^{6} a b^{3} c d + 11 \, \sqrt {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 )}^{6} a^{2} b^{2} d^{2}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} 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 )}^{4}\right )}^{2} a c^{4} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/x^3/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/3*sqrt(b*x + a)*((8*b^4*c^5*d^4*abs(b) - 9*a*b^3*c^4*d^5*abs(b))*(b*x + a)/(b^3*c^9*d - a*b^2*c^8*d^2) + 9*(
b^5*c^6*d^3*abs(b) - 2*a*b^4*c^5*d^4*abs(b) + a^2*b^3*c^4*d^5*abs(b))/(b^3*c^9*d - a*b^2*c^8*d^2))/(b^2*c + (b
*x + a)*b*d - a*b*d)^(3/2) + 1/4*(sqrt(b*d)*b^4*c^2 + 10*sqrt(b*d)*a*b^3*c*d - 35*sqrt(b*d)*a^2*b^2*d^2)*arcta
n(-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)*a*b*c^4*abs(b)) - 1/2*(sqrt(b*d)*b^10*c^5 - 15*sqrt(b*d)*a*b^9*c^4*d + 50*sqrt(b*d)*a^2*b^8*c
^3*d^2 - 70*sqrt(b*d)*a^3*b^7*c^2*d^3 + 45*sqrt(b*d)*a^4*b^6*c*d^4 - 11*sqrt(b*d)*a^5*b^5*d^5 - 3*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^8*c^4 + 40*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^7*c^3*d - 38*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^2*b^6*c^2*d^2 - 32*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2*a^3*b^5*c*d^3 + 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^
4*b^4*d^4 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^6*c^3 - 31*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^5*c^2*d - 19*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^4*c*d^2 - 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^3*d^3 - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^6*b^4*c^2 + 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^3
*c*d + 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^2*d^2)/((b^4*c^2 -
2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4)^2*a*c^4*abs(b))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(1/2)/(x^3*(c + d*x)^(5/2)),x)

[Out]

int((a + b*x)^(1/2)/(x^3*(c + d*x)^(5/2)), x)

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