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

Optimal. Leaf size=204 \[ \frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{3/2}}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}{96 a^2 c x^2}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 a^3 c x}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)}{24 a c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4} \]

[Out]

(-5*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^3*c*x) + (5*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2))/
(96*a^2*c*x^2) - ((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a*c*x^3) - (Sqrt[a + b*x]*(c + d*x)^(7/2))/(4
*c*x^4) + (5*(b*c - a*d)^4*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(3/2))

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Rubi [A]  time = 0.10964, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ \frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{3/2}}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}{96 a^2 c x^2}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 a^3 c x}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)}{24 a c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^3*c*x) + (5*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2))/
(96*a^2*c*x^2) - ((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a*c*x^3) - (Sqrt[a + b*x]*(c + d*x)^(7/2))/(4
*c*x^4) + (5*(b*c - a*d)^4*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(3/2))

Rule 94

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[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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} (c+d x)^{5/2}}{x^5} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}+\frac{(b c-a d) \int \frac{(c+d x)^{5/2}}{x^4 \sqrt{a+b x}} \, dx}{8 c}\\ &=-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}-\frac{\left (5 (b c-a d)^2\right ) \int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx}{48 a c}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/2}}{96 a^2 c x^2}-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}+\frac{\left (5 (b c-a d)^3\right ) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{64 a^2 c}\\ &=-\frac{5 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 a^3 c x}+\frac{5 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/2}}{96 a^2 c x^2}-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}-\frac{\left (5 (b c-a d)^4\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a^3 c}\\ &=-\frac{5 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 a^3 c x}+\frac{5 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/2}}{96 a^2 c x^2}-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}-\frac{\left (5 (b c-a d)^4\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}\\ &=-\frac{5 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 a^3 c x}+\frac{5 (b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/2}}{96 a^2 c x^2}-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}+\frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.363863, size = 176, normalized size = 0.86 \[ \frac{\frac{x (b c-a d) \left (\frac{5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}-8 \sqrt{a+b x} (c+d x)^{5/2}\right )}{a}-48 \sqrt{a+b x} (c+d x)^{7/2}}{192 c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-48*Sqrt[a + b*x]*(c + d*x)^(7/2) + ((b*c - a*d)*x*(-8*Sqrt[a + b*x]*(c + d*x)^(5/2) + (5*(b*c - a*d)*x*(Sqrt
[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqr
t[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(5/2)*Sqrt[c])))/a)/(192*c*x^4)

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Maple [B]  time = 0.015, size = 705, normalized size = 3.5 \begin{align*}{\frac{1}{384\,{a}^{3}c{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}+90\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}{d}^{3}-146\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}bc{d}^{2}+110\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{2}{c}^{2}d-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{b}^{3}{c}^{3}-236\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}c{d}^{2}-72\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{c}^{2}d+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}-272\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{c}^{3}-96\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{c}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*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*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^4*a^3*b*c*d^3+9
0*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^4*a^2*b^2*c^2*d^2-60*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*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
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^4*b^4*c^4-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*d^3
-146*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b*c*d^2+110*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*x^3*a*b^2*c^2*d-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*b^3*c^3-236*(a*c)^(1/2)*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*x^2*a^3*c*d^2-72*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b*c^2*d+20*(a*c)^(1/2)*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a*b^2*c^3-272*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*c^2*d-16*(a*
c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*c^3-96*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*c^3*(a*c)^(1/2))/(
b*d*x^2+a*d*x+b*c*x+a*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((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 28.1691, size = 1247, normalized size = 6.11 \begin{align*} \left [\frac{15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 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} - 55 \, a^{2} b^{2} c^{3} d + 73 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} c^{4} - 18 \, a^{3} b c^{3} d - 59 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (a^{3} b c^{4} + 17 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, a^{4} c^{2} x^{4}}, -\frac{15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 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} - 55 \, a^{2} b^{2} c^{3} d + 73 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} c^{4} - 18 \, a^{3} b c^{3} d - 59 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (a^{3} b c^{4} + 17 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, a^{4} c^{2} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 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 - 55*a^2*b^2*c^3*d + 73*a^3*b*c^2*d^2 + 15*a^4*c*d
^3)*x^3 - 2*(5*a^2*b^2*c^4 - 18*a^3*b*c^3*d - 59*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + 17*a^4*c^3*d)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(a^4*c^2*x^4), -1/384*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 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 - 55*a^2*b^2*c^3*d + 73*a^3*b*c^2*d^2 + 15*
a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 18*a^3*b*c^3*d - 59*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + 17*a^4*c^3*d)*x)*sqr
t(b*x + a)*sqrt(d*x + c))/(a^4*c^2*x^4)]

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

[Out]

Exception raised: TypeError