3.705 \(\int \frac{\sqrt{c+d x}}{x^4 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=191 \[ \frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-3 a d) (a d+3 b c)}{24 a^3 c^2 x}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d)}{12 a^2 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*x^3) + ((5*b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(12*a^2*c*x^2) - ((5*b*
c - 3*a*d)*(3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^3*c^2*x) + ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + a
^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(5/2))

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Rubi [A]  time = 0.139438, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-3 a d) (a d+3 b c)}{24 a^3 c^2 x}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d)}{12 a^2 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(x^4*Sqrt[a + b*x]),x]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*x^3) + ((5*b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(12*a^2*c*x^2) - ((5*b*
c - 3*a*d)*(3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^3*c^2*x) + ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + a
^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(5/2))

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{c+d x}}{x^4 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3}+\frac{\int \frac{\frac{1}{2} (-5 b c+a d)-2 b d x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 a}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3}+\frac{(5 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c x^2}-\frac{\int \frac{-\frac{1}{4} (5 b c-3 a d) (3 b c+a d)-\frac{1}{2} b d (5 b c-a d) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{6 a^2 c}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3}+\frac{(5 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c x^2}-\frac{(5 b c-3 a d) (3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^2 x}+\frac{\int -\frac{3 (b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )}{8 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{6 a^3 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3}+\frac{(5 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c x^2}-\frac{(5 b c-3 a d) (3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^2 x}-\frac{\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 a^3 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3}+\frac{(5 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c x^2}-\frac{(5 b c-3 a d) (3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^2 x}-\frac{\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+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 )}{8 a^3 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3}+\frac{(5 b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c x^2}-\frac{(5 b c-3 a d) (3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^2 x}+\frac{(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.1362, size = 160, normalized size = 0.84 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )+2 a b c x (5 c+2 d x)-15 b^2 c^2 x^2\right )}{24 a^3 c^2 x^3}-\frac{\left (a^2 b c d^2+a^3 d^3+3 a b^2 c^2 d-5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(x^4*Sqrt[a + b*x]),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-15*b^2*c^2*x^2 + 2*a*b*c*x*(5*c + 2*d*x) + a^2*(-8*c^2 - 2*c*d*x + 3*d^2*x^2)))
/(24*a^3*c^2*x^3) - ((-5*b^3*c^3 + 3*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqr
t[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(5/2))

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Maple [B]  time = 0.022, size = 408, normalized size = 2.1 \begin{align*} -{\frac{1}{48\,{a}^{3}{c}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{x}^{2}{a}^{2}{d}^{2}-8\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{x}^{2}abcd+30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{x}^{2}{b}^{2}{c}^{2}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}x{a}^{2}cd-20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xab{c}^{2}+16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{a}^{2}{c}^{2} \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((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),x)

[Out]

-1/48*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^3/c^2*(3*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
^3*a^3*d^3+3*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^3*a^2*b*c*d^2+9*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^3*a*b^2*c^2*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^3*b^3*c^3-6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a^2*d^2-8*((b*x+a)*(d*x+c))^(1/2)*
(a*c)^(1/2)*x^2*a*b*c*d+30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*b^2*c^2+4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/
2)*x*a^2*c*d-20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*a*b*c^2+16*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*c^2)/
((b*x+a)*(d*x+c))^(1/2)/x^3/(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((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.1086, size = 965, normalized size = 5.05 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a c} x^{3} \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 (8 \, a^{3} c^{3} +{\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \,{\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \, a^{4} c^{3} x^{3}}, -\frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-a c} x^{3} \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 (8 \, a^{3} c^{3} +{\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \,{\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \, a^{4} c^{3} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/96*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*sqrt(a*c)*x^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) + 4*(8*a^3*c^3 + (15*a*b^2*c^3 - 4*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 - 2*(5*a^2*b*c^3 - a^3*c^2*d)*x)*sqrt
(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^3), -1/48*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*sqrt(-a*c
)*x^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)) + 2*(8*a^3*c^3 + (15*a*b^2*c^3 - 4*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 - 2*(5*a^2*b*c^3 - a^3*c^
2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^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((d*x+c)**(1/2)/x**4/(b*x+a)**(1/2),x)

[Out]

Timed out

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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((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError