3.478 \(\int \frac{1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=277 \[ -\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{7/2}}-\frac{b (2 b c-a d)}{a^2 c (a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}} \]

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (b*(2*b*c - a*d))/(a^2*c*
(b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - 1/(a*c*x*(a + b*x)*(c + d*x)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a*b
*c*d + 5*a^2*d^2))/(a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a
^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*(b*c - a*d)^(7/2
))

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Rubi [A]  time = 0.426716, antiderivative size = 277, 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 = {103, 151, 152, 156, 63, 208} \[ -\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{7/2}}-\frac{b (2 b c-a d)}{a^2 c (a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^2*(c + d*x)^(5/2)),x]

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (b*(2*b*c - a*d))/(a^2*c*
(b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - 1/(a*c*x*(a + b*x)*(c + d*x)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a*b
*c*d + 5*a^2*d^2))/(a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a
^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*(b*c - a*d)^(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 152

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] && IntegersQ[2*m, 2*n, 2*p]

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{1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx &=-\frac{1}{a c x (a+b x) (c+d x)^{3/2}}-\frac{\int \frac{\frac{1}{2} (4 b c+5 a d)+\frac{7 b d x}{2}}{x (a+b x)^2 (c+d x)^{5/2}} \, dx}{a c}\\ &=-\frac{b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}}-\frac{\int \frac{\frac{1}{2} (b c-a d) (4 b c+5 a d)+\frac{5}{2} b d (2 b c-a d) x}{x (a+b x) (c+d x)^{5/2}} \, dx}{a^2 c (b c-a d)}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}}+\frac{2 \int \frac{-\frac{3}{4} (b c-a d)^2 (4 b c+5 a d)-\frac{3}{4} b d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{x (a+b x) (c+d x)^{3/2}} \, dx}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}-\frac{4 \int \frac{\frac{3}{8} (b c-a d)^3 (4 b c+5 a d)+\frac{3}{8} b d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right ) x}{x (a+b x) \sqrt{c+d x}} \, dx}{3 a^2 c^3 (b c-a d)^3}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}+\frac{\left (b^4 (4 b c-9 a d)\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 a^3 (b c-a d)^3}-\frac{(4 b c+5 a d) \int \frac{1}{x \sqrt{c+d x}} \, dx}{2 a^3 c^3}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}+\frac{\left (b^4 (4 b c-9 a d)\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^3 d (b c-a d)^3}-\frac{(4 b c+5 a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{a^3 c^3 d}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}+\frac{(4 b c+5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{7/2}}-\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.109343, size = 170, normalized size = 0.61 \[ \frac{b^2 c^2 x (a+b x) (4 b c-9 a d) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b (c+d x)}{b c-a d}\right )-(b c-a d) \left (x (a+b x) \left (-5 a^2 d^2+a b c d+4 b^2 c^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{d x}{c}+1\right )-3 a c \left (a^2 d+a b (d x-c)-2 b^2 c x\right )\right )}{3 a^3 c^2 x (a+b x) (c+d x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^2*(c + d*x)^(5/2)),x]

[Out]

(b^2*c^2*(4*b*c - 9*a*d)*x*(a + b*x)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x))/(b*c - a*d)] - (b*c - a*d)
*(-3*a*c*(a^2*d - 2*b^2*c*x + a*b*(-c + d*x)) + (4*b^2*c^2 + a*b*c*d - 5*a^2*d^2)*x*(a + b*x)*Hypergeometric2F
1[-3/2, 1, -1/2, 1 + (d*x)/c]))/(3*a^3*c^2*(b*c - a*d)^2*x*(a + b*x)*(c + d*x)^(3/2))

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Maple [A]  time = 0.026, size = 280, normalized size = 1. \begin{align*} -{\frac{1}{{c}^{3}{a}^{2}x}\sqrt{dx+c}}+5\,{\frac{d}{{c}^{7/2}{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+4\,{\frac{b}{{c}^{5/2}{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{2\,{d}^{3}}{3\,{c}^{2} \left ( ad-bc \right ) ^{2}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}-4\,{\frac{{d}^{4}a}{{c}^{3} \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}+8\,{\frac{{d}^{3}b}{{c}^{2} \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}+{\frac{d{b}^{4}}{{a}^{2} \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+9\,{\frac{d{b}^{4}}{{a}^{2} \left ( ad-bc \right ) ^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-4\,{\frac{{b}^{5}c}{{a}^{3} \left ( ad-bc \right ) ^{3}\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(1/x^2/(b*x+a)^2/(d*x+c)^(5/2),x)

[Out]

-1/c^3/a^2*(d*x+c)^(1/2)/x+5*d/c^(7/2)/a^2*arctanh((d*x+c)^(1/2)/c^(1/2))+4/c^(5/2)/a^3*arctanh((d*x+c)^(1/2)/
c^(1/2))*b-2/3*d^3/c^2/(a*d-b*c)^2/(d*x+c)^(3/2)-4*d^4/c^3/(a*d-b*c)^3/(d*x+c)^(1/2)*a+8*d^3/c^2/(a*d-b*c)^3/(
d*x+c)^(1/2)*b+d*b^4/a^2/(a*d-b*c)^3*(d*x+c)^(1/2)/(b*d*x+a*d)+9*d*b^4/a^2/(a*d-b*c)^3/((a*d-b*c)*b)^(1/2)*arc
tan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-4*b^5/a^3/(a*d-b*c)^3/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*
d-b*c)*b)^(1/2))*c

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 25.2426, size = 7775, normalized size = 28.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^3 + (4
*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x)*sqrt(b/(b*c - a*d))*log(
(b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + 3*((4*b^5*c^4*d^2 - 7*a*b
^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^
2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*
d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^
4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(3*a^2*b
^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b
^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c^2*d^4
- 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)
*x)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^4 + (2*a^3*b^4
*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8
*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d
^2 - a^7*c^6*d^3)*x), -1/6*(6*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2
*b^3*c^4*d^3)*x^3 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x)*sq
rt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) - 3*((4*b^5*c^4*d^2 -
7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 -
13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^
3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*
b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(3
*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11
*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c
^2*d^4 - 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^
2*d^4)*x)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^4 + (2*a
^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a^4*b
^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b
*c^7*d^2 - a^7*c^6*d^3)*x), -1/6*(6*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 -
5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 -
5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5
*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(-
c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - 3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^4*c^5*d
^2 - 9*a^2*b^3*c^4*d^3)*x^3 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2*b^3*c^
6*d)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x +
a)) + 2*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c
^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 +
13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3
- 20*a^5*c^2*d^4)*x)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)
*x^4 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*
c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*
d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x), -1/3*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^
4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^3 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2
*b^3*c^6*d)*x)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) + 3*
((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 1
0*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*
c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b
^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c)
+ (3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3
+ 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4
*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^
5*c^2*d^4)*x)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^4 +
(2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a
^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a
^6*b*c^7*d^2 - a^7*c^6*d^3)*x)]

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Sympy [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**2/(b*x+a)**2/(d*x+c)**(5/2),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.24643, size = 635, normalized size = 2.29 \begin{align*} \frac{{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c^{3} d - 2 \, \sqrt{d x + c} b^{4} c^{4} d - 3 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d^{2} + 4 \, \sqrt{d x + c} a b^{3} c^{3} d^{2} + 3 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{3} - 6 \, \sqrt{d x + c} a^{2} b^{2} c^{2} d^{3} -{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b d^{4} + 4 \, \sqrt{d x + c} a^{3} b c d^{4} - \sqrt{d x + c} a^{4} d^{5}}{{\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )}{\left ({\left (d x + c\right )}^{2} b - 2 \,{\left (d x + c\right )} b c + b c^{2} +{\left (d x + c\right )} a d - a c d\right )}} - \frac{2 \,{\left (12 \,{\left (d x + c\right )} b c d^{3} + b c^{2} d^{3} - 6 \,{\left (d x + c\right )} a d^{4} - a c d^{4}\right )}}{3 \,{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} - \frac{{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c
*d^2 - a^6*d^3)*sqrt(-b^2*c + a*b*d)) - (2*(d*x + c)^(3/2)*b^4*c^3*d - 2*sqrt(d*x + c)*b^4*c^4*d - 3*(d*x + c)
^(3/2)*a*b^3*c^2*d^2 + 4*sqrt(d*x + c)*a*b^3*c^3*d^2 + 3*(d*x + c)^(3/2)*a^2*b^2*c*d^3 - 6*sqrt(d*x + c)*a^2*b
^2*c^2*d^3 - (d*x + c)^(3/2)*a^3*b*d^4 + 4*sqrt(d*x + c)*a^3*b*c*d^4 - sqrt(d*x + c)*a^4*d^5)/((a^2*b^3*c^6 -
3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*
c*d)) - 2/3*(12*(d*x + c)*b*c*d^3 + b*c^2*d^3 - 6*(d*x + c)*a*d^4 - a*c*d^4)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2
*b*c^4*d^2 - a^3*c^3*d^3)*(d*x + c)^(3/2)) - (4*b*c + 5*a*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)*c^3)