3.740 \(\int \frac{1}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )}{24 a^3 c^3 x}+\frac{\left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{7/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{12 a^2 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*x^3) + (5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(12*a^2*c^2*x^2) - ((
15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^3*c^3*x) + ((b*c + a*d)*(5*b^2*c^2 -
2*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(7/2))

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Rubi [A]  time = 0.128342, antiderivative size = 198, 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 = {103, 151, 12, 93, 208} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )}{24 a^3 c^3 x}+\frac{\left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{7/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{12 a^2 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a + b*x]*Sqrt[c + d*x]),x]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*x^3) + (5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(12*a^2*c^2*x^2) - ((
15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^3*c^3*x) + ((b*c + a*d)*(5*b^2*c^2 -
2*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(7/2))

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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{1}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3}-\frac{\int \frac{\frac{5}{2} (b c+a d)+2 b d x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 a c}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3}+\frac{5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c^2 x^2}+\frac{\int \frac{\frac{1}{4} \left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right )+\frac{5}{2} b d (b c+a d) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{6 a^2 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3}+\frac{5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c^2 x^2}-\frac{\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^3 x}-\frac{\int \frac{3 (b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )}{8 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{6 a^3 c^3}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3}+\frac{5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c^2 x^2}-\frac{\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^3 x}-\frac{\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 a^3 c^3}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3}+\frac{5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c^2 x^2}-\frac{\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^3 x}-\frac{\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 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^3}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3}+\frac{5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{12 a^2 c^2 x^2}-\frac{\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{24 a^3 c^3 x}+\frac{(b c+a d) \left (5 b^2 c^2-2 a b c d+5 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^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.123269, size = 162, normalized size = 0.82 \[ \frac{\left (3 a^2 b c d^2+5 a^3 d^3+3 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )+2 a b c x (7 d x-5 c)+15 b^2 c^2 x^2\right )}{24 a^3 c^3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a + b*x]*Sqrt[c + d*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 + 7*d*x) + a^2*(8*c^2 - 10*c*d*x + 15*d^2*x^2)
))/(24*a^3*c^3*x^3) + ((5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])
/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(7/2))

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Maple [B]  time = 0.023, size = 408, normalized size = 2.1 \begin{align*}{\frac{1}{48\,{a}^{3}{c}^{3}{x}^{3}} \left ( 15\,\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}+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}^{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}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{x}^{2}{a}^{2}{d}^{2}-28\,\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}+20\,\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 ) \sqrt{dx+c}\sqrt{bx+a}{\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(1/x^4/(b*x+a)^(1/2)/(d*x+c)^(1/2),x)

[Out]

1/48/a^3/c^3*(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*a^3*d^3+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^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^3-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a^2*d^2-28*((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+20*((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)*(d*x+c)^(1/2)*(b*x+a)^(1/
2)/x^3/(a*c)^(1/2)/((b*x+a)*(d*x+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(1/x^4/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.24597, size = 977, normalized size = 4.93 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, 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} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \,{\left (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^{4} x^{3}}, -\frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, 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} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \,{\left (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^{4} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*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 + 14*a^2*b*c^2*d + 15*a^3*c*d^2)*x^2 - 10*(a^2*b*c^3 + a^3*c^2*d)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^3), -1/48*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*s
qrt(-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 + 14*a^2*b*c^2*d + 15*a^3*c*d^2)*x^2 - 10*(a^2*b*c^3
+ a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt{a + b x} \sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)

[Out]

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

[Out]

Exception raised: TypeError