3.477 \(\int \frac{1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{a^2 c^2 \sqrt{c+d x} (b c-a d)^2}-\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{5/2}}-\frac{b (2 b c-a d)}{a^2 c (a+b x) \sqrt{c+d x} (b c-a d)}-\frac{1}{a c x (a+b x) \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.292002, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{a^2 c^2 \sqrt{c+d x} (b c-a d)^2}-\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{5/2}}-\frac{b (2 b c-a d)}{a^2 c (a+b x) \sqrt{c+d x} (b c-a d)}-\frac{1}{a c x (a+b x) \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^2*(c + d*x)^(3/2)),x]

[Out]

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

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

Mathematica [C]  time = 0.10494, size = 166, normalized size = 0.77 \[ \frac{b^2 c^2 x (a+b x) (4 b c-7 a d) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b (c+d x)}{b c-a d}\right )-(a d-b c) \left (x (a+b x) \left (3 a^2 d^2+a b c d-4 b^2 c^2\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x}{c}+1\right )+a c \left (a^2 d+a b (d x-c)-2 b^2 c x\right )\right )}{a^3 c^2 x (a+b x) \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^2*(c + d*x)^(3/2)),x]

[Out]

(b^2*c^2*(4*b*c - 7*a*d)*x*(a + b*x)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x))/(b*c - a*d)] - (-(b*c) + a*
d)*(a*c*(a^2*d - 2*b^2*c*x + a*b*(-c + d*x)) + (-4*b^2*c^2 + a*b*c*d + 3*a^2*d^2)*x*(a + b*x)*Hypergeometric2F
1[-1/2, 1, 1/2, 1 + (d*x)/c]))/(a^3*c^2*(b*c - a*d)^2*x*(a + b*x)*Sqrt[c + d*x])

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Maple [A]  time = 0.021, size = 229, normalized size = 1.1 \begin{align*} -{\frac{1}{{a}^{2}{c}^{2}x}\sqrt{dx+c}}+3\,{\frac{d}{{a}^{2}{c}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+4\,{\frac{b}{{a}^{3}{c}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{{d}^{3}}{{c}^{2} \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}-{\frac{d{b}^{3}}{{a}^{2} \left ( ad-bc \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-7\,{\frac{d{b}^{3}}{{a}^{2} \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{{b}^{4}c}{{a}^{3} \left ( ad-bc \right ) ^{2}\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)^(3/2),x)

[Out]

-1/a^2/c^2*(d*x+c)^(1/2)/x+3*d/a^2/c^(5/2)*arctanh((d*x+c)^(1/2)/c^(1/2))+4/a^3/c^(3/2)*arctanh((d*x+c)^(1/2)/
c^(1/2))*b-2*d^3/c^2/(a*d-b*c)^2/(d*x+c)^(1/2)-d*b^3/a^2/(a*d-b*c)^2*(d*x+c)^(1/2)/(b*d*x+a*d)-7*d*b^3/a^2/(a*
d-b*c)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))+4*b^4/a^3/(a*d-b*c)^2/((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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 8.57277, size = 4633, normalized size = 21.45 \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)^(3/2),x, algorithm="fricas")

[Out]

[-1/2*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4*a*b^3*c
^5 - 7*a^2*b^2*c^4*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)) - ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*
b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^
2 + 3*a^4*c*d^3)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^
4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2
*d^2 + 3*a^4*c*d^3)*x)*sqrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6
- a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -1/2*(2*(
(4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4*a*b^3*c^5 - 7*a
^2*b^2*c^4*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)) -
((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^
2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt
(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3
*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*s
qrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*
c^4*d^2 + a^6*c^3*d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -1/2*(2*((4*b^4*c^3*d - 5*a*b^3*c
^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a
^4*d^4)*x^2 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(-c)*arctan(sqrt(d*x + c)
*sqrt(-c)/c) + ((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4
*a*b^3*c^5 - 7*a^2*b^2*c^4*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)*sq
rt(b/(b*c - a*d)))/(b*x + a)) + 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*
d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(d*x + c))/((a^3
*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d
^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5
- 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*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)) + ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3
+ 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3*c
^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + (a^2*b^2*
c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a
^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*
d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^
6*c^4*d^2)*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)**(3/2),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.24001, size = 456, normalized size = 2.11 \begin{align*} \frac{{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x + c\right )}^{2} b^{3} c^{2} d - 2 \,{\left (d x + c\right )} b^{3} c^{3} d - 2 \,{\left (d x + c\right )}^{2} a b^{2} c d^{2} + 3 \,{\left (d x + c\right )} a b^{2} c^{2} d^{2} + 3 \,{\left (d x + c\right )}^{2} a^{2} b d^{3} - 7 \,{\left (d x + c\right )} a^{2} b c d^{3} + 2 \, a^{2} b c^{2} d^{3} + 3 \,{\left (d x + c\right )} a^{3} d^{4} - 2 \, a^{3} c d^{4}}{{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )}{\left ({\left (d x + c\right )}^{\frac{5}{2}} b - 2 \,{\left (d x + c\right )}^{\frac{3}{2}} b c + \sqrt{d x + c} b c^{2} +{\left (d x + c\right )}^{\frac{3}{2}} a d - \sqrt{d x + c} a c d\right )}} - \frac{{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^2/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

(4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*sqrt
(-b^2*c + a*b*d)) - (2*(d*x + c)^2*b^3*c^2*d - 2*(d*x + c)*b^3*c^3*d - 2*(d*x + c)^2*a*b^2*c*d^2 + 3*(d*x + c)
*a*b^2*c^2*d^2 + 3*(d*x + c)^2*a^2*b*d^3 - 7*(d*x + c)*a^2*b*c*d^3 + 2*a^2*b*c^2*d^3 + 3*(d*x + c)*a^3*d^4 - 2
*a^3*c*d^4)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*((d*x + c)^(5/2)*b - 2*(d*x + c)^(3/2)*b*c + sqrt(d*x
+ c)*b*c^2 + (d*x + c)^(3/2)*a*d - sqrt(d*x + c)*a*c*d)) - (4*b*c + 3*a*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^
3*sqrt(-c)*c^2)