3.9 \(\int \frac{A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=495 \[ \frac{\sqrt{c+d x} \left (3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right )\right )}{64 b^3 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left (3 a^2 b d (56 c D+C d)-59 a^3 d^2 D-a b^2 \left (-5 B d^2+144 c^2 D+16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )}{96 b^3 (a+b x)^2 (b c-a d)^3}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right )\right )}{64 b^{7/2} (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x} \left (3 a^2 b (8 c D+3 C d)-17 a^3 d D-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{24 b^3 (a+b x)^3 (b c-a d)^2} \]

[Out]

-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)*(a + b*x)^4) - ((b^3*(8*B*c - 7*A*d) -
a*b^2*(16*c*C + B*d) - 17*a^3*d*D + 3*a^2*b*(3*C*d + 8*c*D))*Sqrt[c + d*x])/(24*b^3*(b*c - a*d)^2*(a + b*x)^3
) - ((b^3*(48*c^2*C - 40*B*c*d + 35*A*d^2) - 59*a^3*d^2*D + 3*a^2*b*d*(C*d + 56*c*D) - a*b^2*(16*c*C*d - 5*B*d
^2 + 144*c^2*D))*Sqrt[c + d*x])/(96*b^3*(b*c - a*d)^3*(a + b*x)^2) + ((5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D)
- a*b^2*d*(16*c*C*d - 5*B*d^2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*Sqrt[c + d*x
])/(64*b^3*(b*c - a*d)^4*(a + b*x)) - (d*(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*d*(16*c*C*d - 5*B*d^
2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c
- a*d]])/(64*b^(7/2)*(b*c - a*d)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.00916, antiderivative size = 495, 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, 1157, 385, 199, 208} \[ \frac{\sqrt{c+d x} \left (3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right )\right )}{64 b^3 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left (3 a^2 b d (56 c D+C d)-59 a^3 d^2 D-a b^2 \left (-5 B d^2+144 c^2 D+16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )}{96 b^3 (a+b x)^2 (b c-a d)^3}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right )\right )}{64 b^{7/2} (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x} \left (3 a^2 b (8 c D+3 C d)-17 a^3 d D-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{24 b^3 (a+b x)^3 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^5*Sqrt[c + d*x]),x]

[Out]

-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)*(a + b*x)^4) - ((b^3*(8*B*c - 7*A*d) -
a*b^2*(16*c*C + B*d) - 17*a^3*d*D + 3*a^2*b*(3*C*d + 8*c*D))*Sqrt[c + d*x])/(24*b^3*(b*c - a*d)^2*(a + b*x)^3
) - ((b^3*(48*c^2*C - 40*B*c*d + 35*A*d^2) - 59*a^3*d^2*D + 3*a^2*b*d*(C*d + 56*c*D) - a*b^2*(16*c*C*d - 5*B*d
^2 + 144*c^2*D))*Sqrt[c + d*x])/(96*b^3*(b*c - a*d)^3*(a + b*x)^2) + ((5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D)
- a*b^2*d*(16*c*C*d - 5*B*d^2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*Sqrt[c + d*x
])/(64*b^3*(b*c - a*d)^4*(a + b*x)) - (d*(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*d*(16*c*C*d - 5*B*d^
2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c
- a*d]])/(64*b^(7/2)*(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 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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)^5 \sqrt{c+d x}} \, dx &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt{c+d x}}{4 b^3 (b c-a d) (a+b x)^4}-\frac{\int \frac{-\frac{b^3 (8 B c-7 A d)-a b^2 (8 c C+B d)-a^3 d D+a^2 b (C d+8 c D)}{2 b^3}-\frac{4 (b c-a d) (b C-a D) x}{b^2}-4 \left (c-\frac{a d}{b}\right ) D x^2}{(a+b x)^4 \sqrt{c+d x}} \, dx}{4 (b c-a d)}\\ &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt{c+d x}}{4 b^3 (b c-a d) (a+b x)^4}-\frac{\operatorname{Subst}\left (\int \frac{\frac{-4 c^2 \left (c-\frac{a d}{b}\right ) D+\frac{4 c d (b c-a d) (b C-a D)}{b^2}-\frac{d^2 \left (b^3 (8 B c-7 A d)-a b^2 (8 c C+B d)-a^3 d D+a^2 b (C d+8 c D)\right )}{2 b^3}}{d^2}-\frac{\left (-8 c \left (c-\frac{a d}{b}\right ) D+\frac{4 d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac{4 \left (c-\frac{a d}{b}\right ) D x^4}{d^2}}{\left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )^4} \, dx,x,\sqrt{c+d x}\right )}{2 d (b c-a d)}\\ &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt{c+d x}}{4 b^3 (b c-a d) (a+b x)^4}-\frac{\left (b^3 (8 B c-7 A d)-a b^2 (16 c C+B d)-17 a^3 d D+3 a^2 b (3 C d+8 c D)\right ) \sqrt{c+d x}}{24 b^3 (b c-a d)^2 (a+b x)^3}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (40 B c-\frac{48 c^2 C}{d}-35 A d+\frac{a (16 c C-5 B d)}{b}+\frac{48 c^3 D}{d^2}+\frac{11 a^3 d D}{b^3}-\frac{3 a^2 (C d+8 c D)}{b^2}\right )-\frac{24 (b c-a d)^2 D x^2}{b^2 d^2}}{\left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )^3} \, dx,x,\sqrt{c+d x}\right )}{12 (b c-a d)^2}\\ &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt{c+d x}}{4 b^3 (b c-a d) (a+b x)^4}-\frac{\left (b^3 (8 B c-7 A d)-a b^2 (16 c C+B d)-17 a^3 d D+3 a^2 b (3 C d+8 c D)\right ) \sqrt{c+d x}}{24 b^3 (b c-a d)^2 (a+b x)^3}-\frac{\left (b^3 \left (48 c^2 C-40 B c d+35 A d^2\right )-59 a^3 d^2 D+3 a^2 b d (C d+56 c D)-a b^2 \left (16 c C d-5 B d^2+144 c^2 D\right )\right ) \sqrt{c+d x}}{96 b^3 (b c-a d)^3 (a+b x)^2}-\frac{\left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (16 c C d-5 B d^2-48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 c^3 D\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )^2} \, dx,x,\sqrt{c+d x}\right )}{32 b^3 d (b c-a d)^3}\\ &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt{c+d x}}{4 b^3 (b c-a d) (a+b x)^4}-\frac{\left (b^3 (8 B c-7 A d)-a b^2 (16 c C+B d)-17 a^3 d D+3 a^2 b (3 C d+8 c D)\right ) \sqrt{c+d x}}{24 b^3 (b c-a d)^2 (a+b x)^3}-\frac{\left (b^3 \left (48 c^2 C-40 B c d+35 A d^2\right )-59 a^3 d^2 D+3 a^2 b d (C d+56 c D)-a b^2 \left (16 c C d-5 B d^2+144 c^2 D\right )\right ) \sqrt{c+d x}}{96 b^3 (b c-a d)^3 (a+b x)^2}+\frac{\left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (16 c C d-5 B d^2-48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 c^3 D\right )\right ) \sqrt{c+d x}}{64 b^3 (b c-a d)^4 (a+b x)}+\frac{\left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (16 c C d-5 B d^2-48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 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 )}{64 b^3 (b c-a d)^4}\\ &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt{c+d x}}{4 b^3 (b c-a d) (a+b x)^4}-\frac{\left (b^3 (8 B c-7 A d)-a b^2 (16 c C+B d)-17 a^3 d D+3 a^2 b (3 C d+8 c D)\right ) \sqrt{c+d x}}{24 b^3 (b c-a d)^2 (a+b x)^3}-\frac{\left (b^3 \left (48 c^2 C-40 B c d+35 A d^2\right )-59 a^3 d^2 D+3 a^2 b d (C d+56 c D)-a b^2 \left (16 c C d-5 B d^2+144 c^2 D\right )\right ) \sqrt{c+d x}}{96 b^3 (b c-a d)^3 (a+b x)^2}+\frac{\left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (16 c C d-5 B d^2-48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 c^3 D\right )\right ) \sqrt{c+d x}}{64 b^3 (b c-a d)^4 (a+b x)}-\frac{d \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (16 c C d-5 B d^2-48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 c^3 D\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{7/2} (b c-a d)^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.2285, size = 621, normalized size = 1.25 \[ \frac{7 d (a+b x) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \left (8 \sqrt{b} \sqrt{c+d x} (b c-a d)^{5/2}-5 d (a+b x) \left (3 d^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )+\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d} (-5 a d+2 b c-3 b d x)\right )\right )-48 \sqrt{b} \sqrt{c+d x} (b c-a d)^{7/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )+40 d (a+b x)^2 (b c-a d) \left (3 a^2 D-2 a b C+b^2 B\right ) \left (3 d^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )+\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d} (-5 a d+2 b c-3 b d x)\right )-64 \sqrt{b} (a+b x) \sqrt{c+d x} (b c-a d)^{7/2} \left (3 a^2 D-2 a b C+b^2 B\right )-96 \sqrt{b} (a+b x)^2 \sqrt{c+d x} (b c-a d)^{7/2} (b C-3 a D)+144 d (a+b x)^3 (b c-a d)^2 (b C-3 a D) \left (\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}-d (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )\right )-192 \sqrt{b} D (a+b x)^3 \sqrt{c+d x} (b c-a d)^{7/2}+192 d D (a+b x)^4 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{192 b^{7/2} (a+b x)^4 (b c-a d)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^5*Sqrt[c + d*x]),x]

[Out]

(-48*Sqrt[b]*(b*c - a*d)^(7/2)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x] - 64*Sqrt[b]*(b*c - a*d)^(7/2
)*(b^2*B - 2*a*b*C + 3*a^2*D)*(a + b*x)*Sqrt[c + d*x] - 96*Sqrt[b]*(b*c - a*d)^(7/2)*(b*C - 3*a*D)*(a + b*x)^2
*Sqrt[c + d*x] - 192*Sqrt[b]*(b*c - a*d)^(7/2)*D*(a + b*x)^3*Sqrt[c + d*x] + 192*d*(b*c - a*d)^3*D*(a + b*x)^4
*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]] + 144*d*(b*c - a*d)^2*(b*C - 3*a*D)*(a + b*x)^3*(Sqrt[b]*Sqr
t[b*c - a*d]*Sqrt[c + d*x] - d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]) + 40*d*(b*c - a*d)*
(b^2*B - 2*a*b*C + 3*a^2*D)*(a + b*x)^2*(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]]) + 7*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(a
+ b*x)*(8*Sqrt[b]*(b*c - a*d)^(5/2)*Sqrt[c + d*x] - 5*d*(a + b*x)*(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]])))/(192*b^(7/2)*(b*c
- a*d)^(9/2)*(a + b*x)^4)

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Maple [B]  time = 0.024, size = 3252, normalized size = 6.6 \begin{align*} \text{output 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)^5/(d*x+c)^(1/2),x)

[Out]

-d/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*D*c^3*b^3-3*d/(
b*d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*D*c^3-55/24/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a^2*b*c*d^2+
3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*B*c*d^3+11/4/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c
^3)*(d*x+c)^(5/2)*C*c^2*d^2-73/192/(b*d*x+a*d)^4/b/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)
*a^3*d^4*D-3*d/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*D*c^3+3/64/b^2/(a
^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((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^4-73/24/(b*d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*B*c*d^3-3/8/(b*d*x+a
*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*D*a^2*b*c*d^3+3/4/(b*d*x+a
*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*D*a*b^2*c^2*d^2-11/12/(b*d
*x+a*d)^4*b/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*C*a*c*d^3+3/4/(b*d*x+a*d)^4*b/(a^3*d^3
-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*D*a*c^2*d^2+11/8/(b*d*x+a*d)^4/b/(a^2*d^2-2*a*b*c*d+b^2*c^
2)*(d*x+c)^(3/2)*D*a^2*c*d^3-1/4/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)
*(d*x+c)^(7/2)*C*a*b^2*c*d^3-3/8/b^2/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*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^2*c*d^3+1/4/(b*d*x+a*d)^4/(a*d-b*c)/b*(d*x+c)^(1/2)
*C*a*c*d^3+3/8/(b*d*x+a*d)^4/(a*d-b*c)/b^2*(d*x+c)^(1/2)*D*a^2*c*d^3+3/4/b/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^
2*d^2-4*a*b^3*c^3*d+b^4*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^2-1/4/b
/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*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^3-3/4/(b*d*x+a*d)^4/(a*d-b*c)/b*(d*x+c)^(1/2)*D*a*c^2*d^2+11/64/(b*d*x+a*d)^4/(a^3*
d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*a^2*C*d^4+35/64/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*
a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*A*b^3*d^4+5/64/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2
*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*a^3*d^4*D-d/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*D*c^3+385/
192/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*A*d^4-11/8/(b*d*x+a*d)^4/(a*
d-b*c)*(d*x+c)^(1/2)*B*c*d^3+5/4/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*C*c^2*d^2-5/8/(a^4*d^4-4*a^3*b*c*d^3+6*
a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*B*c*d^3
+3/4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((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^2+73/192/(b*d*x+a*d)^4/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*a*B*d^4-d/(a^4
*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b
*c)*b)^(1/2))*D*c^3+511/192/(b*d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*A*d^4+35/64/(a^4*d^4-4*a
^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(
1/2))*A*d^4-5/64/(b*d*x+a*d)^4/(a*d-b*c)/b*(d*x+c)^(1/2)*a*B*d^4-11/64/(b*d*x+a*d)^4/b/(a^2*d^2-2*a*b*c*d+b^2*
c^2)*(d*x+c)^(3/2)*a^2*C*d^4+3/64/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4
)*(d*x+c)^(7/2)*a^2*b*C*d^4+3/4/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*
(d*x+c)^(7/2)*C*c^2*d^2*b^3+55/192/(b*d*x+a*d)^4*b/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)
*a*B*d^4-55/192/(b*d*x+a*d)^4/b^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*a^3*d^4*D-5/8/(b*d*x+a*d)^4/(a^4*d
^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*B*c*d^3*b^3-5/12/(b*d*x+a*d)^4/(a^2*d^
2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*C*a*c*d^3-3/4/(b*d*x+a*d)^4/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*D*a*c
^2*d^2+13/4/(b*d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*C*c^2*d^2+5/64/b^3/(a^4*d^4-4*a^3*b*c*d^
3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*a^3
*d^4*D-5/64/(b*d*x+a*d)^4/(a*d-b*c)/b^3*(d*x+c)^(1/2)*a^3*d^4*D+5/64/(b*d*x+a*d)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^
2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(d*x+c)^(7/2)*a*b^2*B*d^4-3/64/(b*d*x+a*d)^4/(a*d-b*c)/b^2*(d*x+c)^(1/2)*
a^2*C*d^4+5/64/b/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*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^4+93/64/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*A*d^4+5/8/(b*d*x+a*d)^4
/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)^(5/2)*D*a^2*c*d^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^3+C*x^2+B*x+A)/(b*x+a)^5/(d*x+c)^(1/2),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)^5/(d*x+c)^(1/2),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)**5/(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.56709, size = 2041, normalized size = 4.12 \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)^5/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-1/64*(64*D*b^3*c^3*d - 48*D*a*b^2*c^2*d^2 - 48*C*b^3*c^2*d^2 + 24*D*a^2*b*c*d^3 + 16*C*a*b^2*c*d^3 + 40*B*b^3
*c*d^3 - 5*D*a^3*d^4 - 3*C*a^2*b*d^4 - 5*B*a*b^2*d^4 - 35*A*b^3*d^4)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*
d))/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*sqrt(-b^2*c + a*b*d)) - 1/1
92*(192*(d*x + c)^(7/2)*D*b^6*c^3*d - 576*(d*x + c)^(5/2)*D*b^6*c^4*d + 576*(d*x + c)^(3/2)*D*b^6*c^5*d - 192*
sqrt(d*x + c)*D*b^6*c^6*d - 144*(d*x + c)^(7/2)*D*a*b^5*c^2*d^2 - 144*(d*x + c)^(7/2)*C*b^6*c^2*d^2 + 720*(d*x
+ c)^(5/2)*D*a*b^5*c^3*d^2 + 528*(d*x + c)^(5/2)*C*b^6*c^3*d^2 - 1008*(d*x + c)^(3/2)*D*a*b^5*c^4*d^2 - 624*(
d*x + c)^(3/2)*C*b^6*c^4*d^2 + 432*sqrt(d*x + c)*D*a*b^5*c^5*d^2 + 240*sqrt(d*x + c)*C*b^6*c^5*d^2 + 72*(d*x +
c)^(7/2)*D*a^2*b^4*c*d^3 + 48*(d*x + c)^(7/2)*C*a*b^5*c*d^3 + 120*(d*x + c)^(7/2)*B*b^6*c*d^3 - 24*(d*x + c)^
(5/2)*D*a^2*b^4*c^2*d^3 - 704*(d*x + c)^(5/2)*C*a*b^5*c^2*d^3 - 440*(d*x + c)^(5/2)*B*b^6*c^2*d^3 + 24*(d*x +
c)^(3/2)*D*a^2*b^4*c^3*d^3 + 1328*(d*x + c)^(3/2)*C*a*b^5*c^3*d^3 + 584*(d*x + c)^(3/2)*B*b^6*c^3*d^3 - 72*sqr
t(d*x + c)*D*a^2*b^4*c^4*d^3 - 672*sqrt(d*x + c)*C*a*b^5*c^4*d^3 - 264*sqrt(d*x + c)*B*b^6*c^4*d^3 - 15*(d*x +
c)^(7/2)*D*a^3*b^3*d^4 - 9*(d*x + c)^(7/2)*C*a^2*b^4*d^4 - 15*(d*x + c)^(7/2)*B*a*b^5*d^4 - 105*(d*x + c)^(7/
2)*A*b^6*d^4 - 193*(d*x + c)^(5/2)*D*a^3*b^3*c*d^4 + 209*(d*x + c)^(5/2)*C*a^2*b^4*c*d^4 + 495*(d*x + c)^(5/2)
*B*a*b^5*c*d^4 + 385*(d*x + c)^(5/2)*A*b^6*c*d^4 + 727*(d*x + c)^(3/2)*D*a^3*b^3*c^2*d^4 - 751*(d*x + c)^(3/2)
*C*a^2*b^4*c^2*d^4 - 1241*(d*x + c)^(3/2)*B*a*b^5*c^2*d^4 - 511*(d*x + c)^(3/2)*A*b^6*c^2*d^4 - 471*sqrt(d*x +
c)*D*a^3*b^3*c^3*d^4 + 567*sqrt(d*x + c)*C*a^2*b^4*c^3*d^4 + 777*sqrt(d*x + c)*B*a*b^5*c^3*d^4 + 279*sqrt(d*x
+ c)*A*b^6*c^3*d^4 + 73*(d*x + c)^(5/2)*D*a^4*b^2*d^5 - 33*(d*x + c)^(5/2)*C*a^3*b^3*d^5 - 55*(d*x + c)^(5/2)
*B*a^2*b^4*d^5 - 385*(d*x + c)^(5/2)*A*a*b^5*d^5 - 374*(d*x + c)^(3/2)*D*a^4*b^2*c*d^5 + 14*(d*x + c)^(3/2)*C*
a^3*b^3*c*d^5 + 730*(d*x + c)^(3/2)*B*a^2*b^4*c*d^5 + 1022*(d*x + c)^(3/2)*A*a*b^5*c*d^5 + 405*sqrt(d*x + c)*D
*a^4*b^2*c^2*d^5 - 69*sqrt(d*x + c)*C*a^3*b^3*c^2*d^5 - 747*sqrt(d*x + c)*B*a^2*b^4*c^2*d^5 - 837*sqrt(d*x + c
)*A*a*b^5*c^2*d^5 + 55*(d*x + c)^(3/2)*D*a^5*b*d^6 + 33*(d*x + c)^(3/2)*C*a^4*b^2*d^6 - 73*(d*x + c)^(3/2)*B*a
^3*b^3*d^6 - 511*(d*x + c)^(3/2)*A*a^2*b^4*d^6 - 117*sqrt(d*x + c)*D*a^5*b*c*d^6 - 75*sqrt(d*x + c)*C*a^4*b^2*
c*d^6 + 219*sqrt(d*x + c)*B*a^3*b^3*c*d^6 + 837*sqrt(d*x + c)*A*a^2*b^4*c*d^6 + 15*sqrt(d*x + c)*D*a^6*d^7 + 9
*sqrt(d*x + c)*C*a^5*b*d^7 + 15*sqrt(d*x + c)*B*a^4*b^2*d^7 - 279*sqrt(d*x + c)*A*a^3*b^3*d^7)/((b^7*c^4 - 4*a
*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*((d*x + c)*b - b*c + a*d)^4)