3.475 \(\int \frac{(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=284 \[ -\frac{b \sqrt{c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{8 a^4 (a+b x)}-\frac{\sqrt{c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{24 a^3 x (a+b x)}+\frac{\left (60 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{8 a^5 \sqrt{c}}+\frac{c \sqrt{c+d x} (8 b c-9 a d)}{12 a^2 x^2 (a+b x)}-\frac{\sqrt{b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^5}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)} \]

[Out]

-(b*(32*b^2*c^2 - 52*a*b*c*d + 19*a^2*d^2)*Sqrt[c + d*x])/(8*a^4*(a + b*x)) + (c*(8*b*c - 9*a*d)*Sqrt[c + d*x]
)/(12*a^2*x^2*(a + b*x)) - ((48*b^2*c^2 - 82*a*b*c*d + 33*a^2*d^2)*Sqrt[c + d*x])/(24*a^3*x*(a + b*x)) - (c*(c
+ d*x)^(3/2))/(3*a*x^3*(a + b*x)) + ((64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[Sqrt
[c + d*x]/Sqrt[c]])/(8*a^5*Sqrt[c]) - (Sqrt[b]*(8*b*c - 3*a*d)*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x
])/Sqrt[b*c - a*d]])/a^5

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Rubi [A]  time = 0.396916, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {98, 149, 151, 156, 63, 208} \[ -\frac{b \sqrt{c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{8 a^4 (a+b x)}-\frac{\sqrt{c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{24 a^3 x (a+b x)}+\frac{\left (60 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{8 a^5 \sqrt{c}}+\frac{c \sqrt{c+d x} (8 b c-9 a d)}{12 a^2 x^2 (a+b x)}-\frac{\sqrt{b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^5}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]

[Out]

-(b*(32*b^2*c^2 - 52*a*b*c*d + 19*a^2*d^2)*Sqrt[c + d*x])/(8*a^4*(a + b*x)) + (c*(8*b*c - 9*a*d)*Sqrt[c + d*x]
)/(12*a^2*x^2*(a + b*x)) - ((48*b^2*c^2 - 82*a*b*c*d + 33*a^2*d^2)*Sqrt[c + d*x])/(24*a^3*x*(a + b*x)) - (c*(c
+ d*x)^(3/2))/(3*a*x^3*(a + b*x)) + ((64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[Sqrt
[c + d*x]/Sqrt[c]])/(8*a^5*Sqrt[c]) - (Sqrt[b]*(8*b*c - 3*a*d)*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x
])/Sqrt[b*c - a*d]])/a^5

Rule 98

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

Rule 149

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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{(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx &=-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)}-\frac{\int \frac{\sqrt{c+d x} \left (\frac{1}{2} c (8 b c-9 a d)+\frac{1}{2} d (5 b c-6 a d) x\right )}{x^3 (a+b x)^2} \, dx}{3 a}\\ &=\frac{c (8 b c-9 a d) \sqrt{c+d x}}{12 a^2 x^2 (a+b x)}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)}-\frac{\int \frac{-\frac{1}{4} c \left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right )-\frac{1}{4} d \left (40 b^2 c^2-65 a b c d+24 a^2 d^2\right ) x}{x^2 (a+b x)^2 \sqrt{c+d x}} \, dx}{6 a^2}\\ &=\frac{c (8 b c-9 a d) \sqrt{c+d x}}{12 a^2 x^2 (a+b x)}-\frac{\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt{c+d x}}{24 a^3 x (a+b x)}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac{\int \frac{-\frac{3}{8} c \left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right )-\frac{3}{8} b c d \left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) x}{x (a+b x)^2 \sqrt{c+d x}} \, dx}{6 a^3 c}\\ &=-\frac{b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt{c+d x}}{8 a^4 (a+b x)}+\frac{c (8 b c-9 a d) \sqrt{c+d x}}{12 a^2 x^2 (a+b x)}-\frac{\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt{c+d x}}{24 a^3 x (a+b x)}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac{\int \frac{-\frac{3}{8} c (b c-a d) \left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right )-\frac{3}{8} b c d (b c-a d) \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) x}{x (a+b x) \sqrt{c+d x}} \, dx}{6 a^4 c (b c-a d)}\\ &=-\frac{b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt{c+d x}}{8 a^4 (a+b x)}+\frac{c (8 b c-9 a d) \sqrt{c+d x}}{12 a^2 x^2 (a+b x)}-\frac{\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt{c+d x}}{24 a^3 x (a+b x)}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac{\left (b (8 b c-3 a d) (b c-a d)^2\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 a^5}-\frac{\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \int \frac{1}{x \sqrt{c+d x}} \, dx}{16 a^5}\\ &=-\frac{b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt{c+d x}}{8 a^4 (a+b x)}+\frac{c (8 b c-9 a d) \sqrt{c+d x}}{12 a^2 x^2 (a+b x)}-\frac{\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt{c+d x}}{24 a^3 x (a+b x)}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac{\left (b (8 b c-3 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{a^5 d}-\frac{\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{8 a^5 d}\\ &=-\frac{b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt{c+d x}}{8 a^4 (a+b x)}+\frac{c (8 b c-9 a d) \sqrt{c+d x}}{12 a^2 x^2 (a+b x)}-\frac{\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt{c+d x}}{24 a^3 x (a+b x)}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac{\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{8 a^5 \sqrt{c}}-\frac{\sqrt{b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.535444, size = 237, normalized size = 0.83 \[ -\frac{\frac{a \sqrt{c+d x} \left (a^2 b x \left (-16 c^2-82 c d x+57 d^2 x^2\right )+a^3 \left (8 c^2+26 c d x+33 d^2 x^2\right )+12 a b^2 c x^2 (4 c-13 d x)+96 b^3 c^2 x^3\right )}{x^3 (a+b x)}-\frac{3 \left (60 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c}}+24 \sqrt{b} \sqrt{b c-a d} \left (3 a^2 d^2-11 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{24 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]

[Out]

-((a*Sqrt[c + d*x]*(96*b^3*c^2*x^3 + 12*a*b^2*c*x^2*(4*c - 13*d*x) + a^3*(8*c^2 + 26*c*d*x + 33*d^2*x^2) + a^2
*b*x*(-16*c^2 - 82*c*d*x + 57*d^2*x^2)))/(x^3*(a + b*x)) - (3*(64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 -
5*a^3*d^3)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/Sqrt[c] + 24*Sqrt[b]*Sqrt[b*c - a*d]*(8*b^2*c^2 - 11*a*b*c*d + 3*a
^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(24*a^5)

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Maple [B]  time = 0.017, size = 545, normalized size = 1.9 \begin{align*} -{\frac{11}{8\,{a}^{2}{x}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{9\,bc}{2\,d{a}^{3}{x}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-3\,{\frac{ \left ( dx+c \right ) ^{5/2}{b}^{2}{c}^{2}}{{d}^{2}{a}^{4}{x}^{3}}}+{\frac{5\,c}{3\,{a}^{2}{x}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-8\,{\frac{ \left ( dx+c \right ) ^{3/2}b{c}^{2}}{d{a}^{3}{x}^{3}}}+6\,{\frac{ \left ( dx+c \right ) ^{3/2}{b}^{2}{c}^{3}}{{d}^{2}{a}^{4}{x}^{3}}}+{\frac{7\,b{c}^{3}}{2\,d{a}^{3}{x}^{3}}\sqrt{dx+c}}-3\,{\frac{{b}^{2}\sqrt{dx+c}{c}^{4}}{{d}^{2}{a}^{4}{x}^{3}}}-{\frac{5\,{c}^{2}}{8\,{a}^{2}{x}^{3}}\sqrt{dx+c}}-{\frac{5\,{d}^{3}}{8\,{a}^{2}}{\it Artanh} \left ({\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+{\frac{15\,{d}^{2}b}{2\,{a}^{3}}\sqrt{c}{\it Artanh} \left ({\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }-15\,{\frac{d{c}^{3/2}{b}^{2}}{{a}^{4}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+8\,{\frac{{c}^{5/2}{b}^{3}}{{a}^{5}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{{d}^{3}b}{{a}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+2\,{\frac{{d}^{2}{b}^{2}\sqrt{dx+c}c}{{a}^{3} \left ( bdx+ad \right ) }}-{\frac{d{b}^{3}{c}^{2}}{{a}^{4} \left ( bdx+ad \right ) }\sqrt{dx+c}}-3\,{\frac{{d}^{3}b}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+14\,{\frac{{d}^{2}{b}^{2}c}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-19\,{\frac{d{b}^{3}{c}^{2}}{{a}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+8\,{\frac{{b}^{4}{c}^{3}}{{a}^{5}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/x^4/(b*x+a)^2,x)

[Out]

-11/8/a^2/x^3*(d*x+c)^(5/2)+9/2/d/a^3/x^3*(d*x+c)^(5/2)*c*b-3/d^2/a^4/x^3*(d*x+c)^(5/2)*b^2*c^2+5/3/a^2/x^3*(d
*x+c)^(3/2)*c-8/d/a^3/x^3*(d*x+c)^(3/2)*b*c^2+6/d^2/a^4/x^3*(d*x+c)^(3/2)*b^2*c^3+7/2/d/a^3/x^3*(d*x+c)^(1/2)*
b*c^3-3/d^2/a^4/x^3*(d*x+c)^(1/2)*b^2*c^4-5/8/a^2/x^3*(d*x+c)^(1/2)*c^2-5/8*d^3/a^2/c^(1/2)*arctanh((d*x+c)^(1
/2)/c^(1/2))+15/2*d^2/a^3*c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b-15*d/a^4*c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1
/2))*b^2+8/a^5*c^(5/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b^3-d^3*b/a^2*(d*x+c)^(1/2)/(b*d*x+a*d)+2*d^2*b^2/a^3*(d
*x+c)^(1/2)/(b*d*x+a*d)*c-d*b^3/a^4*(d*x+c)^(1/2)/(b*d*x+a*d)*c^2-3*d^3*b/a^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*
x+c)^(1/2)/((a*d-b*c)*b)^(1/2))+14*d^2*b^2/a^3/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))
*c-19*d*b^3/a^4/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*c^2+8*b^4/a^5/((a*d-b*c)*b)^(1
/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*c^3

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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)/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.46188, size = 3225, normalized size = 11.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

[1/48*(24*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c*d^2)*x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3
)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 3*((64*b^4*
c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*b^2*c^2*d + 60*a^3*b*c*d
^2 - 5*a^4*d^3)*x^3)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(8*a^4*c^3 + 3*(32*a*b^3*c^3 - 5
2*a^2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3
- 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3), 1/48*(48*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c
*d^2)*x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)
*sqrt(d*x + c)/(b*d*x + b*c)) - 3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a
*b^3*c^3 - 120*a^2*b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2
*c)/x) - 2*(8*a^4*c^3 + 3*(32*a*b^3*c^3 - 52*a^2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*
c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3), -1/24*
(3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*b^2*c^2*d +
60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - 12*((8*b^3*c^3 - 11*a*b^2*c^2*d +
3*a^2*b*c*d^2)*x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c
- a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + (8*a^4*c^3 + 3*(32*a*b^3*c^3 - 52*a^2*b^2*c^2*d + 1
9*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3 - 13*a^4*c^2*d)*x)*
sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3), 1/24*(24*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c*d^2)*x^4 + (8*a*b^
2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d
*x + b*c)) - 3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*
b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - (8*a^4*c^3 + 3*(32*a*
b^3*c^3 - 52*a^2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8
*a^3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*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)**(5/2)/x**4/(b*x+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.2416, size = 500, normalized size = 1.76 \begin{align*} \frac{{\left (8 \, b^{4} c^{3} - 19 \, a b^{3} c^{2} d + 14 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{5}} - \frac{{\left (64 \, b^{3} c^{3} - 120 \, a b^{2} c^{2} d + 60 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{8 \, a^{5} \sqrt{-c}} - \frac{\sqrt{d x + c} b^{3} c^{2} d - 2 \, \sqrt{d x + c} a b^{2} c d^{2} + \sqrt{d x + c} a^{2} b d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a^{4}} - \frac{72 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} c^{2} d - 144 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c^{3} d + 72 \, \sqrt{d x + c} b^{2} c^{4} d - 108 \,{\left (d x + c\right )}^{\frac{5}{2}} a b c d^{2} + 192 \,{\left (d x + c\right )}^{\frac{3}{2}} a b c^{2} d^{2} - 84 \, \sqrt{d x + c} a b c^{3} d^{2} + 33 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} d^{3} - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} c d^{3} + 15 \, \sqrt{d x + c} a^{2} c^{2} d^{3}}{24 \, a^{4} d^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^2,x, algorithm="giac")

[Out]

(8*b^4*c^3 - 19*a*b^3*c^2*d + 14*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sq
rt(-b^2*c + a*b*d)*a^5) - 1/8*(64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*arctan(sqrt(d*x + c)
/sqrt(-c))/(a^5*sqrt(-c)) - (sqrt(d*x + c)*b^3*c^2*d - 2*sqrt(d*x + c)*a*b^2*c*d^2 + sqrt(d*x + c)*a^2*b*d^3)/
(((d*x + c)*b - b*c + a*d)*a^4) - 1/24*(72*(d*x + c)^(5/2)*b^2*c^2*d - 144*(d*x + c)^(3/2)*b^2*c^3*d + 72*sqrt
(d*x + c)*b^2*c^4*d - 108*(d*x + c)^(5/2)*a*b*c*d^2 + 192*(d*x + c)^(3/2)*a*b*c^2*d^2 - 84*sqrt(d*x + c)*a*b*c
^3*d^2 + 33*(d*x + c)^(5/2)*a^2*d^3 - 40*(d*x + c)^(3/2)*a^2*c*d^3 + 15*sqrt(d*x + c)*a^2*c^2*d^3)/(a^4*d^3*x^
3)