3.584 \(\int \frac{\sqrt{a+b x}}{x^5 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=279 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+6 a b c d+5 b^2 c^2\right )}{96 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (25 a^2 b c d^2-105 a^3 d^3+17 a b^2 c^2 d+15 b^3 c^3\right )}{192 a^3 c^4 x}+\frac{(b c-a d) \left (15 a^2 b c d^2+35 a^3 d^3+9 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-7 a d)}{24 a c^2 x^3}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c*x^4) - ((b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a*c^2*x^3) + ((5*b^
2*c^2 + 6*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a^2*c^3*x^2) - ((15*b^3*c^3 + 17*a*b^2*c^2*d
+ 25*a^2*b*c*d^2 - 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^3*c^4*x) + ((b*c - a*d)*(5*b^3*c^3 + 9*a*b
^2*c^2*d + 15*a^2*b*c*d^2 + 35*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*
c^(9/2))

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Rubi [A]  time = 0.225177, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {99, 151, 12, 93, 208} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+6 a b c d+5 b^2 c^2\right )}{96 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (25 a^2 b c d^2-105 a^3 d^3+17 a b^2 c^2 d+15 b^3 c^3\right )}{192 a^3 c^4 x}+\frac{(b c-a d) \left (15 a^2 b c d^2+35 a^3 d^3+9 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-7 a d)}{24 a c^2 x^3}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c*x^4) - ((b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a*c^2*x^3) + ((5*b^
2*c^2 + 6*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a^2*c^3*x^2) - ((15*b^3*c^3 + 17*a*b^2*c^2*d
+ 25*a^2*b*c*d^2 - 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^3*c^4*x) + ((b*c - a*d)*(5*b^3*c^3 + 9*a*b
^2*c^2*d + 15*a^2*b*c*d^2 + 35*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*
c^(9/2))

Rule 99

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 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}}{x^5 \sqrt{c+d x}} \, dx &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}+\frac{\int \frac{\frac{1}{2} (b c-7 a d)-3 b d x}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{4 c}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c^2 x^3}-\frac{\int \frac{\frac{1}{4} \left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right )+b d (b c-7 a d) x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{12 a c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c^2 x^3}+\frac{\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^3 x^2}+\frac{\int \frac{\frac{1}{8} \left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right )+\frac{1}{4} b d \left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^2 c^3}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c^2 x^3}+\frac{\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^3 x^2}-\frac{\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^4 x}-\frac{\int \frac{3 (b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right )}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^3 c^4}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c^2 x^3}+\frac{\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^3 x^2}-\frac{\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^4 x}-\frac{\left ((b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a^3 c^4}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c^2 x^3}+\frac{\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^3 x^2}-\frac{\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^4 x}-\frac{\left ((b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 a^3 c^4}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c^2 x^3}+\frac{\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^3 x^2}-\frac{\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^4 x}+\frac{(b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.258469, size = 244, normalized size = 0.87 \[ \frac{-\frac{2 x^2 (a+b x)^{3/2} \sqrt{c+d x} \left (35 a^2 d^2+22 a b c d+15 b^2 c^2\right )}{a^2 c^2}+\frac{3 x^3 \left (15 a^2 b c d^2+35 a^3 d^3+9 a b^2 c^2 d+5 b^3 c^3\right ) \left (\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+x (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )}{a^{5/2} c^{7/2}}+\frac{8 x (a+b x)^{3/2} \sqrt{c+d x} (7 a d+5 b c)}{a c}-48 (a+b x)^{3/2} \sqrt{c+d x}}{192 a c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-48*(a + b*x)^(3/2)*Sqrt[c + d*x] + (8*(5*b*c + 7*a*d)*x*(a + b*x)^(3/2)*Sqrt[c + d*x])/(a*c) - (2*(15*b^2*c^
2 + 22*a*b*c*d + 35*a^2*d^2)*x^2*(a + b*x)^(3/2)*Sqrt[c + d*x])/(a^2*c^2) + (3*(5*b^3*c^3 + 9*a*b^2*c^2*d + 15
*a^2*b*c*d^2 + 35*a^3*d^3)*x^3*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x] + (b*c - a*d)*x*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(5/2)*c^(7/2)))/(192*a*c*x^4)

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Maple [B]  time = 0.023, size = 593, normalized size = 2.1 \begin{align*} -{\frac{1}{384\,{a}^{3}{c}^{4}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-210\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{3}{d}^{3}+50\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{2}bc{d}^{2}+34\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}a{b}^{2}{c}^{2}d+30\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{b}^{3}{c}^{3}+140\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{3}c{d}^{2}-24\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{2}b{c}^{2}d-20\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}a{b}^{2}{c}^{3}-112\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{3}{c}^{2}d+16\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{c}^{3}+96\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{c}^{3} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x
)*x^4*a^4*d^4-60*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^3*b*c*d^3-18*ln((a*d*x+
b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^2*b^2*c^2*d^2-12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a*b^3*c^3*d-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)
/x)*x^4*b^4*c^4-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^3*d^3+50*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3
*a^2*b*c*d^2+34*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a*b^2*c^2*d+30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3
*b^3*c^3+140*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^3*c*d^2-24*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^2*
b*c^2*d-20*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a*b^2*c^3-112*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^3*c^2
*d+16*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b*c^3+96*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c^3)/((b*x+a)
*(d*x+c))^(1/2)/x^4/(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)^(1/2)/x^5/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 28.5813, size = 1260, normalized size = 4.52 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \sqrt{a c} x^{4} \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 (48 \, a^{4} c^{4} +{\left (15 \, a b^{3} c^{4} + 17 \, a^{2} b^{2} c^{3} d + 25 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} c^{4} + 6 \, a^{3} b c^{3} d - 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, a^{4} c^{5} x^{4}}, -\frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \sqrt{-a c} x^{4} \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 (48 \, a^{4} c^{4} +{\left (15 \, a b^{3} c^{4} + 17 \, a^{2} b^{2} c^{3} d + 25 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} c^{4} + 6 \, a^{3} b c^{3} d - 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, a^{4} c^{5} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 + 17*a^2*b^2*c^3*d + 25*a^3*b*c^2*d^2 - 105*
a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 + 6*a^3*b*c^3*d - 35*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 - 7*a^4*c^3*d)*x)*sqrt(
b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^4), -1/384*(3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d
^3 - 35*a^4*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 + 17*a^2*b^2*c^3*d + 25*a^3*b*c^2
*d^2 - 105*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 + 6*a^3*b*c^3*d - 35*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 - 7*a^4*c^3*
d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x}}{x^{5} \sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError