3.17 \(\int \frac{A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=463 \[ -\frac{\sqrt{c+d x} \left (-a^2 b d (11 C d-18 c D)+5 a^3 d^2 D+a b^2 \left (-7 B d^2-72 c^2 D+36 c C d\right )+b^3 \left (49 A d^2-42 B c d+24 c^2 C\right )\right )}{24 b^2 (a+b x) (b c-a d)^4}+\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left (-\left (7 A d^3-6 B c d^2+6 c^2 C d-6 c^3 D\right )\right )}{3 b^3 \sqrt{c+d x} (b c-a d)^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^2 b d^2 (C d-6 c D)+a^3 d^3 D-a b^2 d \left (-5 B d^2-24 c^2 D+12 c C d\right )+b^3 \left (-\left (35 A d^3-30 B c d^2+24 c^2 C d-16 c^3 D\right )\right )\right )}{8 b^{5/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{c+d x} \left (a^2 b (18 c D+5 C d)-11 a^3 d D-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{12 b^2 (a+b x)^2 (b c-a d)^3} \]

[Out]

(a*b^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(6*c^2*C*d - 6*B*c*d^2 + 7*A*d^3 - 6*c^3*D))/(3*b^3*(b*c - a*d)^4
*Sqrt[c + d*x]) - (A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(3*b^3*(b*c - a*d)*(a + b*x)^3*Sqrt[c + d*x]) - ((b^3*(6
*B*c - 7*A*d) - a*b^2*(12*c*C - B*d) - 11*a^3*d*D + a^2*b*(5*C*d + 18*c*D))*Sqrt[c + d*x])/(12*b^2*(b*c - a*d)
^3*(a + b*x)^2) - ((b^3*(24*c^2*C - 42*B*c*d + 49*A*d^2) + 5*a^3*d^2*D - a^2*b*d*(11*C*d - 18*c*D) + a*b^2*(36
*c*C*d - 7*B*d^2 - 72*c^2*D))*Sqrt[c + d*x])/(24*b^2*(b*c - a*d)^4*(a + b*x)) - ((a^3*d^3*D + a^2*b*d^2*(C*d -
6*c*D) - a*b^2*d*(12*c*C*d - 5*B*d^2 - 24*c^2*D) - b^3*(24*c^2*C*d - 30*B*c*d^2 + 35*A*d^3 - 16*c^3*D))*ArcTa
nh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(8*b^(5/2)*(b*c - a*d)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.14999, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1621, 897, 1259, 456, 453, 208} \[ -\frac{\sqrt{c+d x} \left (-a^2 b d (11 C d-18 c D)+5 a^3 d^2 D+a b^2 \left (-7 B d^2-72 c^2 D+36 c C d\right )+b^3 \left (49 A d^2-42 B c d+24 c^2 C\right )\right )}{24 b^2 (a+b x) (b c-a d)^4}+\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left (-\left (7 A d^3-6 B c d^2+6 c^2 C d-6 c^3 D\right )\right )}{3 b^3 \sqrt{c+d x} (b c-a d)^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (a^2 b d^2 (C d-6 c D)+a^3 d^3 D-a b^2 d \left (-5 B d^2-24 c^2 D+12 c C d\right )+b^3 \left (-\left (35 A d^3-30 B c d^2+24 c^2 C d-16 c^3 D\right )\right )\right )}{8 b^{5/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{c+d x} \left (a^2 b (18 c D+5 C d)-11 a^3 d D-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{12 b^2 (a+b x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*(c + d*x)^(3/2)),x]

[Out]

(a*b^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(6*c^2*C*d - 6*B*c*d^2 + 7*A*d^3 - 6*c^3*D))/(3*b^3*(b*c - a*d)^4
*Sqrt[c + d*x]) - (A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(3*b^3*(b*c - a*d)*(a + b*x)^3*Sqrt[c + d*x]) - ((b^3*(6
*B*c - 7*A*d) - a*b^2*(12*c*C - B*d) - 11*a^3*d*D + a^2*b*(5*C*d + 18*c*D))*Sqrt[c + d*x])/(12*b^2*(b*c - a*d)
^3*(a + b*x)^2) - ((b^3*(24*c^2*C - 42*B*c*d + 49*A*d^2) + 5*a^3*d^2*D - a^2*b*d*(11*C*d - 18*c*D) + a*b^2*(36
*c*C*d - 7*B*d^2 - 72*c^2*D))*Sqrt[c + d*x])/(24*b^2*(b*c - a*d)^4*(a + b*x)) - ((a^3*d^3*D + a^2*b*d^2*(C*d -
6*c*D) - a*b^2*d*(12*c*C*d - 5*B*d^2 - 24*c^2*D) - b^3*(24*c^2*C*d - 30*B*c*d^2 + 35*A*d^3 - 16*c^3*D))*ArcTa
nh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(8*b^(5/2)*(b*c - a*d)^(9/2))

Rule 1621

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px,
a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((m + 1)*
(b*c - a*d)), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c -
a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expo
n[Px, x], 2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^3 (b c-a d) (a+b x)^3 \sqrt{c+d x}}-\frac{\int \frac{-\frac{b^3 (6 B c-7 A d)-a b^2 (6 c C-B d)+a^3 d D-a^2 b (C d-6 c D)}{2 b^3}-\frac{3 (b c-a d) (b C-a D) x}{b^2}-3 \left (c-\frac{a d}{b}\right ) D x^2}{(a+b x)^3 (c+d x)^{3/2}} \, dx}{3 (b c-a d)}\\ &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^3 (b c-a d) (a+b x)^3 \sqrt{c+d x}}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{-3 c^2 \left (c-\frac{a d}{b}\right ) D+\frac{3 c d (b c-a d) (b C-a D)}{b^2}-\frac{d^2 \left (b^3 (6 B c-7 A d)-a b^2 (6 c C-B d)+a^3 d D-a^2 b (C d-6 c D)\right )}{2 b^3}}{d^2}-\frac{\left (-6 c \left (c-\frac{a d}{b}\right ) D+\frac{3 d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac{3 \left (c-\frac{a d}{b}\right ) D x^4}{d^2}}{x^2 \left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )^3} \, dx,x,\sqrt{c+d x}\right )}{3 d (b c-a d)}\\ &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^3 (b c-a d) (a+b x)^3 \sqrt{c+d x}}-\frac{\left (b^3 (6 B c-7 A d)-a b^2 (12 c C-B d)-11 a^3 d D+a^2 b (5 C d+18 c D)\right ) \sqrt{c+d x}}{12 b^2 (b c-a d)^3 (a+b x)^2}+\frac{d^3 \operatorname{Subst}\left (\int \frac{-\frac{2 (b c-a d) \left (a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (6 c^2 C d-6 B c d^2+7 A d^3-6 c^3 D\right )\right )}{b d^5}+\frac{3 \left (3 a^3 d^3 D-a^2 b d^2 (5 C d-6 c D)+a b^2 d \left (12 c C d-B d^2-24 c^2 D\right )-b^3 \left (6 B c d^2-7 A d^3-8 c^3 D\right )\right ) x^2}{2 d^5}}{x^2 \left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )^2} \, dx,x,\sqrt{c+d x}\right )}{6 b^2 (b c-a d)^3}\\ &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^3 (b c-a d) (a+b x)^3 \sqrt{c+d x}}-\frac{\left (b^3 (6 B c-7 A d)-a b^2 (12 c C-B d)-11 a^3 d D+a^2 b (5 C d+18 c D)\right ) \sqrt{c+d x}}{12 b^2 (b c-a d)^3 (a+b x)^2}-\frac{\left (b^3 \left (24 c^2 C-42 B c d+49 A d^2\right )+5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (36 c C d-7 B d^2-72 c^2 D\right )\right ) \sqrt{c+d x}}{24 b^2 (b c-a d)^4 (a+b x)}-\frac{d^3 \operatorname{Subst}\left (\int \frac{-\frac{4 \left (a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (6 c^2 C d-6 B c d^2+7 A d^3-6 c^3 D\right )\right )}{b d^4}+\frac{\left (b^3 \left (24 c^2 C-42 B c d+49 A d^2\right )+5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (36 c C d-7 B d^2-72 c^2 D\right )\right ) x^2}{2 d^3 (b c-a d)}}{x^2 \left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )} \, dx,x,\sqrt{c+d x}\right )}{12 b^2 (b c-a d)^3}\\ &=\frac{a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (6 c^2 C d-6 B c d^2+7 A d^3-6 c^3 D\right )}{3 b^3 (b c-a d)^4 \sqrt{c+d x}}-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^3 (b c-a d) (a+b x)^3 \sqrt{c+d x}}-\frac{\left (b^3 (6 B c-7 A d)-a b^2 (12 c C-B d)-11 a^3 d D+a^2 b (5 C d+18 c D)\right ) \sqrt{c+d x}}{12 b^2 (b c-a d)^3 (a+b x)^2}-\frac{\left (b^3 \left (24 c^2 C-42 B c d+49 A d^2\right )+5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (36 c C d-7 B d^2-72 c^2 D\right )\right ) \sqrt{c+d x}}{24 b^2 (b c-a d)^4 (a+b x)}+\frac{\left (a^3 d^3 D+a^2 b d^2 (C d-6 c D)-a b^2 d \left (12 c C d-5 B d^2-24 c^2 D\right )-b^3 \left (24 c^2 C d-30 B c d^2+35 A d^3-16 c^3 D\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{-b c+a d}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{8 b^2 d (b c-a d)^4}\\ &=\frac{a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (6 c^2 C d-6 B c d^2+7 A d^3-6 c^3 D\right )}{3 b^3 (b c-a d)^4 \sqrt{c+d x}}-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^3 (b c-a d) (a+b x)^3 \sqrt{c+d x}}-\frac{\left (b^3 (6 B c-7 A d)-a b^2 (12 c C-B d)-11 a^3 d D+a^2 b (5 C d+18 c D)\right ) \sqrt{c+d x}}{12 b^2 (b c-a d)^3 (a+b x)^2}-\frac{\left (b^3 \left (24 c^2 C-42 B c d+49 A d^2\right )+5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (36 c C d-7 B d^2-72 c^2 D\right )\right ) \sqrt{c+d x}}{24 b^2 (b c-a d)^4 (a+b x)}-\frac{\left (a^3 d^3 D+a^2 b d^2 (C d-6 c D)-a b^2 d \left (12 c C d-5 B d^2-24 c^2 D\right )-b^3 \left (24 c^2 C d-30 B c d^2+35 A d^3-16 c^3 D\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.52002, size = 697, normalized size = 1.51 \[ \frac{\sqrt{c+d x} \left (-3 a^2 b c d D+a^3 d^2 D+3 a b^2 c^2 D-b^3 \left (A d^2-B c d+c^2 C\right )\right )}{b^2 (a+b x) (b c-a d)^4}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (3 a^2 b c d D+a^3 \left (-d^2\right ) D-3 a b^2 c^2 D+b^3 \left (A d^2-B c d+c^2 C\right )\right )}{b^{5/2} (b c-a d)^{9/2}}+\frac{5 d \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \left (3 d^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )+\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d} (-5 a d+2 b c-3 b d x)\right )}{24 b^{5/2} (a+b x)^2 (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left (a^2 b (3 c D+C d)-2 a^3 d D-2 a b^2 c C+b^3 (B c-A d)\right )}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac{\sqrt{c+d x} \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{3 b^2 (a+b x)^3 (b c-a d)^2}+\frac{3 d \left (d (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )-\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}\right ) \left (-a^2 b (3 c D+C d)+2 a^3 d D+2 a b^2 c C+b^3 (A d-B c)\right )}{4 b^{5/2} (a+b x) (b c-a d)^{9/2}}+\frac{2 \left (-A d^3+B c d^2-c^2 C d+c^3 D\right )}{\sqrt{c+d x} (b c-a d)^4}+\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{(b c-a d)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*(c + d*x)^(3/2)),x]

[Out]

(2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/((b*c - a*d)^4*Sqrt[c + d*x]) + ((-(A*b^3) + a*(b^2*B - a*b*C + a^2
*D))*Sqrt[c + d*x])/(3*b^2*(b*c - a*d)^2*(a + b*x)^3) - ((-2*a*b^2*c*C + b^3*(B*c - A*d) - 2*a^3*d*D + a^2*b*(
C*d + 3*c*D))*Sqrt[c + d*x])/(2*b^2*(b*c - a*d)^3*(a + b*x)^2) + ((-(b^3*(c^2*C - B*c*d + A*d^2)) + 3*a*b^2*c^
2*D - 3*a^2*b*c*d*D + a^3*d^2*D)*Sqrt[c + d*x])/(b^2*(b*c - a*d)^4*(a + b*x)) + (2*Sqrt[b]*(c^2*C*d - B*c*d^2
+ A*d^3 - c^3*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(9/2) + (d*(b^3*(c^2*C - B*c*d
+ A*d^2) - 3*a*b^2*c^2*D + 3*a^2*b*c*d*D - a^3*d^2*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/
2)*(b*c - a*d)^(9/2)) + (3*d*(2*a*b^2*c*C + b^3*(-(B*c) + A*d) + 2*a^3*d*D - a^2*b*(C*d + 3*c*D))*(-(Sqrt[b]*S
qrt[b*c - a*d]*Sqrt[c + d*x]) + d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(4*b^(5/2)*(b*c
- a*d)^(9/2)*(a + b*x)) + (5*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]*(2*
b*c - 5*a*d - 3*b*d*x) + 3*d^2*(a + b*x)^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(24*b^(5/2)*(b*c
- a*d)^(9/2)*(a + b*x)^2)

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Maple [B]  time = 0.034, size = 2108, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(3/2),x)

[Out]

-3/4/(a*d-b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*D*a^2*b*c*d^2+1/(a*d-b*c)^4/(b*d*x+a*d)^3*d^4/b*(d*x+c)^(1/2)*D*a
^4*c+6/(a*d-b*c)^4/(b*d*x+a*d)^3*b*d^2*(d*x+c)^(3/2)*D*a^2*c^2+3/(a*d-b*c)^4/(b*d*x+a*d)^3*d*b^2*(d*x+c)^(1/2)
*D*a*c^4-21/4/(a*d-b*c)^4/(b*d*x+a*d)^3*d^2*b*(d*x+c)^(1/2)*D*a^2*c^3+29/4/(a*d-b*c)^4/(b*d*x+a*d)^3*d^4*b^2*(
d*x+c)^(1/2)*A*a*c+31/8/(a*d-b*c)^4/(b*d*x+a*d)^3*d^3*b*(d*x+c)^(1/2)*C*a^2*c^2-1/2/(a*d-b*c)^4/(b*d*x+a*d)^3*
d^2*b^2*(d*x+c)^(1/2)*C*a*c^3-3/2/(a*d-b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*C*a*b^2*c*d^2-25/8/(a*d-b*c)^4/(b*d*
x+a*d)^3*d^3*b^2*(d*x+c)^(1/2)*B*a*c^2+1/8/(a*d-b*c)^4/b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)
*b)^(1/2))*a^2*C*d^3+1/8/(a*d-b*c)^4/b^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*a^3*d
^3*D+15/4/(a*d-b*c)^4*b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*B*c*d^2+1/8/(a*d-b*c)^
4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*a^2*b*C*d^3+11/8/(a*d-b*c)^4/(b*d*x+a*d)^3*d^3*(d*x+c)^(1/2)*D*a^3*c^2+1/3/(a*d-
b*c)^4/(b*d*x+a*d)^3*d^3*(d*x+c)^(3/2)*D*a^3*c+7/4/(a*d-b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*B*b^3*c*d^2-1/(a*d-
b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*C*b^3*c^2*d+1/3/(a*d-b*c)^4/(b*d*x+a*d)^3*d^4*(d*x+c)^(3/2)*C*a^3+11/8/(a*d
-b*c)^4/(b*d*x+a*d)^3*d^5*(d*x+c)^(1/2)*B*a^3-19/8/(a*d-b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*A*b^3*d^3+1/8/(a*d-
b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*a^3*d^3*D+5/8/(a*d-b*c)^4/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-
b*c)*b)^(1/2))*a*B*d^3-35/8/(a*d-b*c)^4*b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*A*d^
3+2/(a*d-b*c)^4*b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*D*c^3-2/(a*d-b*c)^4/(d*x+c)^
(1/2)*A*d^3+2/(a*d-b*c)^4/(d*x+c)^(1/2)*D*c^3+3/(a*d-b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*D*a*b^2*c^2*d-3/4/(a*d
-b*c)^4/b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*D*a^2*c*d^2+2/(a*d-b*c)^4/(b*d*x+a*d
)^3*b^2*d^2*(d*x+c)^(3/2)*C*a*c^2-1/2/(a*d-b*c)^4/(b*d*x+a*d)^3*d^4*b*(d*x+c)^(1/2)*B*a^2*c-13/3/(a*d-b*c)^4/(
b*d*x+a*d)^3*b*d^3*(d*x+c)^(3/2)*C*a^2*c+7/3/(a*d-b*c)^4/(b*d*x+a*d)^3*b^2*d^3*(d*x+c)^(3/2)*B*a*c-6/(a*d-b*c)
^4/(b*d*x+a*d)^3*b^2*d*(d*x+c)^(3/2)*D*a*c^3+2/(a*d-b*c)^4/(d*x+c)^(1/2)*B*c*d^2-2/(a*d-b*c)^4/(d*x+c)^(1/2)*C
*c^2*d+5/8/(a*d-b*c)^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)*a*b^2*B*d^3-3/2/(a*d-b*c)^4/((a*d-b*c)*b)^(1/2)*arctan(b*(d
*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*C*a*c*d^2+3/(a*d-b*c)^4/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)
*b)^(1/2))*D*a*c^2*d-9/4/(a*d-b*c)^4/(b*d*x+a*d)^3*d^4*(d*x+c)^(1/2)*C*a^3*c-4/(a*d-b*c)^4/(b*d*x+a*d)^3*b^3*d
^2*(d*x+c)^(3/2)*B*c^2+2/(a*d-b*c)^4/(b*d*x+a*d)^3*b^3*d*(d*x+c)^(3/2)*C*c^3-1/3/(a*d-b*c)^4/(b*d*x+a*d)^3/b*d
^4*(d*x+c)^(3/2)*D*a^4-29/8/(a*d-b*c)^4/(b*d*x+a*d)^3*d^5*b*(d*x+c)^(1/2)*A*a^2-29/8/(a*d-b*c)^4/(b*d*x+a*d)^3
*d^3*b^3*(d*x+c)^(1/2)*A*c^2+9/4/(a*d-b*c)^4/(b*d*x+a*d)^3*d^2*b^3*(d*x+c)^(1/2)*B*c^3-1/8/(a*d-b*c)^4/(b*d*x+
a*d)^3*d^5/b*(d*x+c)^(1/2)*C*a^4-1/(a*d-b*c)^4/(b*d*x+a*d)^3*d*b^3*(d*x+c)^(1/2)*C*c^4-1/8/(a*d-b*c)^4/(b*d*x+
a*d)^3*d^5/b^2*(d*x+c)^(1/2)*D*a^5-17/3/(a*d-b*c)^4/(b*d*x+a*d)^3*b^2*d^4*(d*x+c)^(3/2)*A*a+17/3/(a*d-b*c)^4/(
b*d*x+a*d)^3*b^3*d^3*(d*x+c)^(3/2)*A*c+5/3/(a*d-b*c)^4/(b*d*x+a*d)^3*b*d^4*(d*x+c)^(3/2)*B*a^2-3/(a*d-b*c)^4*b
/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*C*c^2*d

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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^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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**3+C*x**2+B*x+A)/(b*x+a)**4/(d*x+c)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.67914, size = 1465, normalized size = 3.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

1/8*(16*D*b^3*c^3 + 24*D*a*b^2*c^2*d - 24*C*b^3*c^2*d - 6*D*a^2*b*c*d^2 - 12*C*a*b^2*c*d^2 + 30*B*b^3*c*d^2 +
D*a^3*d^3 + C*a^2*b*d^3 + 5*B*a*b^2*d^3 - 35*A*b^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^6*c^4
- 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*sqrt(-b^2*c + a*b*d)) + 2*(D*c^3 - C*c^2
*d + B*c*d^2 - A*d^3)/((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(d*x + c))
+ 1/24*(72*(d*x + c)^(5/2)*D*a*b^4*c^2*d - 24*(d*x + c)^(5/2)*C*b^5*c^2*d - 144*(d*x + c)^(3/2)*D*a*b^4*c^3*d
+ 48*(d*x + c)^(3/2)*C*b^5*c^3*d + 72*sqrt(d*x + c)*D*a*b^4*c^4*d - 24*sqrt(d*x + c)*C*b^5*c^4*d - 18*(d*x + c
)^(5/2)*D*a^2*b^3*c*d^2 - 36*(d*x + c)^(5/2)*C*a*b^4*c*d^2 + 42*(d*x + c)^(5/2)*B*b^5*c*d^2 + 144*(d*x + c)^(3
/2)*D*a^2*b^3*c^2*d^2 + 48*(d*x + c)^(3/2)*C*a*b^4*c^2*d^2 - 96*(d*x + c)^(3/2)*B*b^5*c^2*d^2 - 126*sqrt(d*x +
c)*D*a^2*b^3*c^3*d^2 - 12*sqrt(d*x + c)*C*a*b^4*c^3*d^2 + 54*sqrt(d*x + c)*B*b^5*c^3*d^2 + 3*(d*x + c)^(5/2)*
D*a^3*b^2*d^3 + 3*(d*x + c)^(5/2)*C*a^2*b^3*d^3 + 15*(d*x + c)^(5/2)*B*a*b^4*d^3 - 57*(d*x + c)^(5/2)*A*b^5*d^
3 + 8*(d*x + c)^(3/2)*D*a^3*b^2*c*d^3 - 104*(d*x + c)^(3/2)*C*a^2*b^3*c*d^3 + 56*(d*x + c)^(3/2)*B*a*b^4*c*d^3
+ 136*(d*x + c)^(3/2)*A*b^5*c*d^3 + 33*sqrt(d*x + c)*D*a^3*b^2*c^2*d^3 + 93*sqrt(d*x + c)*C*a^2*b^3*c^2*d^3 -
75*sqrt(d*x + c)*B*a*b^4*c^2*d^3 - 87*sqrt(d*x + c)*A*b^5*c^2*d^3 - 8*(d*x + c)^(3/2)*D*a^4*b*d^4 + 8*(d*x +
c)^(3/2)*C*a^3*b^2*d^4 + 40*(d*x + c)^(3/2)*B*a^2*b^3*d^4 - 136*(d*x + c)^(3/2)*A*a*b^4*d^4 + 24*sqrt(d*x + c)
*D*a^4*b*c*d^4 - 54*sqrt(d*x + c)*C*a^3*b^2*c*d^4 - 12*sqrt(d*x + c)*B*a^2*b^3*c*d^4 + 174*sqrt(d*x + c)*A*a*b
^4*c*d^4 - 3*sqrt(d*x + c)*D*a^5*d^5 - 3*sqrt(d*x + c)*C*a^4*b*d^5 + 33*sqrt(d*x + c)*B*a^3*b^2*d^5 - 87*sqrt(
d*x + c)*A*a^2*b^3*d^5)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*((d*x +
c)*b - b*c + a*d)^3)