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

Optimal. Leaf size=229 \[ \frac{5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{9/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (7 a d+b c)}{24 a c^2 x^3}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d) (7 a d+b c)}{96 a c^3 x^2}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (7 a d+b c)}{64 a c^4 x}-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4} \]

[Out]

(5*(b*c - a*d)^2*(b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a*c^4*x) + (5*(b*c - a*d)*(b*c + 7*a*d)*(a + b
*x)^(3/2)*Sqrt[c + d*x])/(96*a*c^3*x^2) + ((b*c + 7*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*a*c^2*x^3) - ((a +
b*x)^(7/2)*Sqrt[c + d*x])/(4*a*c*x^4) + (5*(b*c - a*d)^3*(b*c + 7*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[
a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(9/2))

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Rubi [A]  time = 0.114745, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ \frac{5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{9/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (7 a d+b c)}{24 a c^2 x^3}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d) (7 a d+b c)}{96 a c^3 x^2}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (7 a d+b c)}{64 a c^4 x}-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b*c - a*d)^2*(b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a*c^4*x) + (5*(b*c - a*d)*(b*c + 7*a*d)*(a + b
*x)^(3/2)*Sqrt[c + d*x])/(96*a*c^3*x^2) + ((b*c + 7*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*a*c^2*x^3) - ((a +
b*x)^(7/2)*Sqrt[c + d*x])/(4*a*c*x^4) + (5*(b*c - a*d)^3*(b*c + 7*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[
a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(9/2))

Rule 96

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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{(a+b x)^{5/2}}{x^5 \sqrt{c+d x}} \, dx &=-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4}-\frac{\left (\frac{b c}{2}+\frac{7 a d}{2}\right ) \int \frac{(a+b x)^{5/2}}{x^4 \sqrt{c+d x}} \, dx}{4 a c}\\ &=\frac{(b c+7 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 a c^2 x^3}-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4}-\frac{(5 (b c-a d) (b c+7 a d)) \int \frac{(a+b x)^{3/2}}{x^3 \sqrt{c+d x}} \, dx}{48 a c^2}\\ &=\frac{5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{(b c+7 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 a c^2 x^3}-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4}-\frac{\left (5 (b c-a d)^2 (b c+7 a d)\right ) \int \frac{\sqrt{a+b x}}{x^2 \sqrt{c+d x}} \, dx}{64 a c^3}\\ &=\frac{5 (b c-a d)^2 (b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^4 x}+\frac{5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{(b c+7 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 a c^2 x^3}-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4}-\frac{\left (5 (b c-a d)^3 (b c+7 a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a c^4}\\ &=\frac{5 (b c-a d)^2 (b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^4 x}+\frac{5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{(b c+7 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 a c^2 x^3}-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4}-\frac{\left (5 (b c-a d)^3 (b c+7 a d)\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 c^4}\\ &=\frac{5 (b c-a d)^2 (b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 a c^4 x}+\frac{5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{(b c+7 a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 a c^2 x^3}-\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 a c x^4}+\frac{5 (b c-a d)^3 (b c+7 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.200329, size = 179, normalized size = 0.78 \[ \frac{\frac{x (7 a d+b c) \left (\frac{5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c-3 a d x+5 b c x)\right )}{\sqrt{a} c^{5/2}}+8 (a+b x)^{5/2} \sqrt{c+d x}\right )}{c}-48 (a+b x)^{7/2} \sqrt{c+d x}}{192 a c x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.022, size = 593, normalized size = 2.6 \begin{align*} -{\frac{1}{384\,a{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}-300\,\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}+270\,\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}-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{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}+530\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{2}bc{d}^{2}-382\,\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}-344\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{2}b{c}^{2}d+236\,\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+272\,\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)^(5/2)/x^5/(d*x+c)^(1/2),x)

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/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^4-300*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+270*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^2-60*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*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+530*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^
3*a^2*b*c*d^2-382*(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^2-344*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a
^2*b*c^2*d+236*(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+272*(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)^(5/2)/x^5/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 27.7321, size = 1276, normalized size = 5.57 \begin{align*} \left [-\frac{15 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, 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} - 191 \, a^{2} b^{2} c^{3} d + 265 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (59 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (17 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, a^{2} c^{5} x^{4}}, -\frac{15 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, 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} - 191 \, a^{2} b^{2} c^{3} d + 265 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (59 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (17 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, a^{2} c^{5} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*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 - 191*a^2*b^2*c^3*d + 265*a^3*b*c^2*d^2 - 105
*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 - 86*a^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 - 7*a^4*c^3*d)*x)
*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^5*x^4), -1/384*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3
*b*c*d^3 - 7*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 - 191*a^2*b^2*c^3*d + 265*a^
3*b*c^2*d^2 - 105*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 - 86*a^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4
- 7*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^5*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((b*x+a)**(5/2)/x**5/(d*x+c)**(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((b*x+a)^(5/2)/x^5/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError