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

[Out]

-1/64*d*(5*a^3*d^3*D+3*a^2*b*d^2*(C*d-8*D*c)-a*b^2*d*(-5*B*d^2+16*C*c*d-48*D*c^2)+b^3*(35*A*d^3-40*B*c*d^2+48*
C*c^2*d-64*D*c^3))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(7/2)/(-a*d+b*c)^(9/2)-1/4*(A*b^3-a*(B*b^
2-C*a*b+D*a^2))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)/(b*x+a)^4-1/24*(b^3*(-7*A*d+8*B*c)-a*b^2*(B*d+16*C*c)-17*a^3*d*D+
3*a^2*b*(3*C*d+8*D*c))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)^2/(b*x+a)^3-1/96*(b^3*(35*A*d^2-40*B*c*d+48*C*c^2)-59*a^3*
d^2*D+3*a^2*b*d*(C*d+56*D*c)-a*b^2*(-5*B*d^2+16*C*c*d+144*D*c^2))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)^3/(b*x+a)^2+1/6
4*(5*a^3*d^3*D+3*a^2*b*d^2*(C*d-8*D*c)-a*b^2*d*(-5*B*d^2+16*C*c*d-48*D*c^2)+b^3*(35*A*d^3-40*B*c*d^2+48*C*c^2*
d-64*D*c^3))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)^4/(b*x+a)

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Rubi [A]  time = 1.01, antiderivative size = 495, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 2.42, 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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fricas [B]  time = 1.21, size = 3624, normalized size = 7.32 \[ \text {result too large to display} \]

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]

[-1/384*(3*(64*D*a^4*b^3*c^3*d - (5*D*a^7 + 3*C*a^6*b + 5*B*a^5*b^2 + 35*A*a^4*b^3)*d^4 + (64*D*b^7*c^3*d - (5
*D*a^3*b^4 + 3*C*a^2*b^5 + 5*B*a*b^6 + 35*A*b^7)*d^4 + 8*(3*D*a^2*b^5*c + (2*C*a*b^6 + 5*B*b^7)*c)*d^3 - 48*(D
*a*b^6*c^2 + C*b^7*c^2)*d^2)*x^4 + 8*(3*D*a^6*b*c + (2*C*a^5*b^2 + 5*B*a^4*b^3)*c)*d^3 + 4*(64*D*a*b^6*c^3*d -
(5*D*a^4*b^3 + 3*C*a^3*b^4 + 5*B*a^2*b^5 + 35*A*a*b^6)*d^4 + 8*(3*D*a^3*b^4*c + (2*C*a^2*b^5 + 5*B*a*b^6)*c)*
d^3 - 48*(D*a^2*b^5*c^2 + C*a*b^6*c^2)*d^2)*x^3 - 48*(D*a^5*b^2*c^2 + C*a^4*b^3*c^2)*d^2 + 6*(64*D*a^2*b^5*c^3
*d - (5*D*a^5*b^2 + 3*C*a^4*b^3 + 5*B*a^3*b^4 + 35*A*a^2*b^5)*d^4 + 8*(3*D*a^4*b^3*c + (2*C*a^3*b^4 + 5*B*a^2*
b^5)*c)*d^3 - 48*(D*a^3*b^4*c^2 + C*a^2*b^5*c^2)*d^2)*x^2 + 4*(64*D*a^3*b^4*c^3*d - (5*D*a^6*b + 3*C*a^5*b^2 +
5*B*a^4*b^3 + 35*A*a^3*b^4)*d^4 + 8*(3*D*a^5*b^2*c + (2*C*a^4*b^3 + 5*B*a^3*b^4)*c)*d^3 - 48*(D*a^4*b^3*c^2 +
C*a^3*b^4*c^2)*d^2)*x)*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)) + 2*(48*D*a^3*b^5*c^4 + 16*(C*a^2*b^6 + B*a*b^7 + 3*A*b^8)*c^4 - 3*(5*D*a^7*b + 3*C*a^6*b^2 + 5*B*a^5
*b^3 - 93*A*a^4*b^4)*d^4 + (77*D*a^6*b^2*c + (51*C*a^5*b^3 - 131*B*a^4*b^4 - 605*A*a^3*b^5)*c)*d^3 + 3*(64*D*b
^8*c^4 + (5*D*a^4*b^4 + 3*C*a^3*b^5 + 5*B*a^2*b^6 + 35*A*a*b^7)*d^4 - (29*D*a^3*b^5*c + (19*C*a^2*b^6 + 45*B*a
*b^7 + 35*A*b^8)*c)*d^3 + 8*(9*D*a^2*b^6*c^2 + (8*C*a*b^7 + 5*B*b^8)*c^2)*d^2 - 16*(7*D*a*b^7*c^3 + 3*C*b^8*c^
3)*d)*x^3 - 2*(83*D*a^5*b^3*c^2 - (23*C*a^4*b^4 + 109*B*a^3*b^5 + 263*A*a^2*b^6)*c^2)*d^2 + (288*D*a*b^7*c^4 +
96*C*b^8*c^4 - (73*D*a^5*b^3 - 33*C*a^4*b^4 - 55*B*a^3*b^5 - 385*A*a^2*b^6)*d^4 + (311*D*a^4*b^4*c - 5*(43*C*
a^3*b^5 + 101*B*a^2*b^6 + 91*A*a*b^7)*c)*d^3 - 2*(215*D*a^3*b^5*c^2 - (371*C*a^2*b^6 + 265*B*a*b^7 + 35*A*b^8)
*c^2)*d^2 - 16*(6*D*a^2*b^6*c^3 + (41*C*a*b^7 + 5*B*b^8)*c^3)*d)*x^2 + 8*(7*D*a^4*b^4*c^3 - (13*C*a^3*b^5 + 11
*B*a^2*b^6 + 31*A*a*b^7)*c^3)*d + (192*D*a^2*b^6*c^4 + 64*(C*a*b^7 + B*b^8)*c^4 - (55*D*a^6*b^2 + 33*C*a^5*b^3
- 73*B*a^4*b^4 - 511*A*a^3*b^5)*d^4 + (283*D*a^5*b^3*c + (85*C*a^4*b^4 - 693*B*a^3*b^5 - 763*A*a^2*b^6)*c)*d^
3 - 4*(131*D*a^4*b^4*c^2 - (77*C*a^3*b^5 + 229*B*a^2*b^6 + 77*A*a*b^7)*c^2)*d^2 + 8*(13*D*a^3*b^5*c^3 - (53*C*
a^2*b^6 + 45*B*a*b^7 + 7*A*b^8)*c^3)*d)*x)*sqrt(d*x + c))/(a^4*b^9*c^5 - 5*a^5*b^8*c^4*d + 10*a^6*b^7*c^3*d^2
- 10*a^7*b^6*c^2*d^3 + 5*a^8*b^5*c*d^4 - a^9*b^4*d^5 + (b^13*c^5 - 5*a*b^12*c^4*d + 10*a^2*b^11*c^3*d^2 - 10*a
^3*b^10*c^2*d^3 + 5*a^4*b^9*c*d^4 - a^5*b^8*d^5)*x^4 + 4*(a*b^12*c^5 - 5*a^2*b^11*c^4*d + 10*a^3*b^10*c^3*d^2
- 10*a^4*b^9*c^2*d^3 + 5*a^5*b^8*c*d^4 - a^6*b^7*d^5)*x^3 + 6*(a^2*b^11*c^5 - 5*a^3*b^10*c^4*d + 10*a^4*b^9*c^
3*d^2 - 10*a^5*b^8*c^2*d^3 + 5*a^6*b^7*c*d^4 - a^7*b^6*d^5)*x^2 + 4*(a^3*b^10*c^5 - 5*a^4*b^9*c^4*d + 10*a^5*b
^8*c^3*d^2 - 10*a^6*b^7*c^2*d^3 + 5*a^7*b^6*c*d^4 - a^8*b^5*d^5)*x), -1/192*(3*(64*D*a^4*b^3*c^3*d - (5*D*a^7
+ 3*C*a^6*b + 5*B*a^5*b^2 + 35*A*a^4*b^3)*d^4 + (64*D*b^7*c^3*d - (5*D*a^3*b^4 + 3*C*a^2*b^5 + 5*B*a*b^6 + 35*
A*b^7)*d^4 + 8*(3*D*a^2*b^5*c + (2*C*a*b^6 + 5*B*b^7)*c)*d^3 - 48*(D*a*b^6*c^2 + C*b^7*c^2)*d^2)*x^4 + 8*(3*D*
a^6*b*c + (2*C*a^5*b^2 + 5*B*a^4*b^3)*c)*d^3 + 4*(64*D*a*b^6*c^3*d - (5*D*a^4*b^3 + 3*C*a^3*b^4 + 5*B*a^2*b^5
+ 35*A*a*b^6)*d^4 + 8*(3*D*a^3*b^4*c + (2*C*a^2*b^5 + 5*B*a*b^6)*c)*d^3 - 48*(D*a^2*b^5*c^2 + C*a*b^6*c^2)*d^2
)*x^3 - 48*(D*a^5*b^2*c^2 + C*a^4*b^3*c^2)*d^2 + 6*(64*D*a^2*b^5*c^3*d - (5*D*a^5*b^2 + 3*C*a^4*b^3 + 5*B*a^3*
b^4 + 35*A*a^2*b^5)*d^4 + 8*(3*D*a^4*b^3*c + (2*C*a^3*b^4 + 5*B*a^2*b^5)*c)*d^3 - 48*(D*a^3*b^4*c^2 + C*a^2*b^
5*c^2)*d^2)*x^2 + 4*(64*D*a^3*b^4*c^3*d - (5*D*a^6*b + 3*C*a^5*b^2 + 5*B*a^4*b^3 + 35*A*a^3*b^4)*d^4 + 8*(3*D*
a^5*b^2*c + (2*C*a^4*b^3 + 5*B*a^3*b^4)*c)*d^3 - 48*(D*a^4*b^3*c^2 + C*a^3*b^4*c^2)*d^2)*x)*sqrt(-b^2*c + a*b*
d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) + (48*D*a^3*b^5*c^4 + 16*(C*a^2*b^6 + B*a*b^7 + 3*
A*b^8)*c^4 - 3*(5*D*a^7*b + 3*C*a^6*b^2 + 5*B*a^5*b^3 - 93*A*a^4*b^4)*d^4 + (77*D*a^6*b^2*c + (51*C*a^5*b^3 -
131*B*a^4*b^4 - 605*A*a^3*b^5)*c)*d^3 + 3*(64*D*b^8*c^4 + (5*D*a^4*b^4 + 3*C*a^3*b^5 + 5*B*a^2*b^6 + 35*A*a*b^
7)*d^4 - (29*D*a^3*b^5*c + (19*C*a^2*b^6 + 45*B*a*b^7 + 35*A*b^8)*c)*d^3 + 8*(9*D*a^2*b^6*c^2 + (8*C*a*b^7 + 5
*B*b^8)*c^2)*d^2 - 16*(7*D*a*b^7*c^3 + 3*C*b^8*c^3)*d)*x^3 - 2*(83*D*a^5*b^3*c^2 - (23*C*a^4*b^4 + 109*B*a^3*b
^5 + 263*A*a^2*b^6)*c^2)*d^2 + (288*D*a*b^7*c^4 + 96*C*b^8*c^4 - (73*D*a^5*b^3 - 33*C*a^4*b^4 - 55*B*a^3*b^5 -
385*A*a^2*b^6)*d^4 + (311*D*a^4*b^4*c - 5*(43*C*a^3*b^5 + 101*B*a^2*b^6 + 91*A*a*b^7)*c)*d^3 - 2*(215*D*a^3*b
^5*c^2 - (371*C*a^2*b^6 + 265*B*a*b^7 + 35*A*b^8)*c^2)*d^2 - 16*(6*D*a^2*b^6*c^3 + (41*C*a*b^7 + 5*B*b^8)*c^3)
*d)*x^2 + 8*(7*D*a^4*b^4*c^3 - (13*C*a^3*b^5 + 11*B*a^2*b^6 + 31*A*a*b^7)*c^3)*d + (192*D*a^2*b^6*c^4 + 64*(C*
a*b^7 + B*b^8)*c^4 - (55*D*a^6*b^2 + 33*C*a^5*b^3 - 73*B*a^4*b^4 - 511*A*a^3*b^5)*d^4 + (283*D*a^5*b^3*c + (85
*C*a^4*b^4 - 693*B*a^3*b^5 - 763*A*a^2*b^6)*c)*d^3 - 4*(131*D*a^4*b^4*c^2 - (77*C*a^3*b^5 + 229*B*a^2*b^6 + 77
*A*a*b^7)*c^2)*d^2 + 8*(13*D*a^3*b^5*c^3 - (53*C*a^2*b^6 + 45*B*a*b^7 + 7*A*b^8)*c^3)*d)*x)*sqrt(d*x + c))/(a^
4*b^9*c^5 - 5*a^5*b^8*c^4*d + 10*a^6*b^7*c^3*d^2 - 10*a^7*b^6*c^2*d^3 + 5*a^8*b^5*c*d^4 - a^9*b^4*d^5 + (b^13*
c^5 - 5*a*b^12*c^4*d + 10*a^2*b^11*c^3*d^2 - 10*a^3*b^10*c^2*d^3 + 5*a^4*b^9*c*d^4 - a^5*b^8*d^5)*x^4 + 4*(a*b
^12*c^5 - 5*a^2*b^11*c^4*d + 10*a^3*b^10*c^3*d^2 - 10*a^4*b^9*c^2*d^3 + 5*a^5*b^8*c*d^4 - a^6*b^7*d^5)*x^3 + 6
*(a^2*b^11*c^5 - 5*a^3*b^10*c^4*d + 10*a^4*b^9*c^3*d^2 - 10*a^5*b^8*c^2*d^3 + 5*a^6*b^7*c*d^4 - a^7*b^6*d^5)*x
^2 + 4*(a^3*b^10*c^5 - 5*a^4*b^9*c^4*d + 10*a^5*b^8*c^3*d^2 - 10*a^6*b^7*c^2*d^3 + 5*a^7*b^6*c*d^4 - a^8*b^5*d
^5)*x)]

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giac [B]  time = 1.65, size = 1512, normalized size = 3.05 \[ \text {result too large to display} \]

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)

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maple [B]  time = 0.03, size = 3252, normalized size = 6.57 \[ \text {output too large to display} \]

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]

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((d*x+c)^(1/2)
/((a*d-b*c)*b)^(1/2)*b)*D*a*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+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-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((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*D*c
^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((d*x+c)^(1/2
)/((a*d-b*c)*b)^(1/2)*b)*C*c^2*d^2-d/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*D*c^3+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+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^4-1/4/(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^3-11/12/(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^3-5/8/(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((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*B*c*d^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+5/4/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*C*c^2*d^2-11/8/(b*d*x+a*d)^4/(a*d-
b*c)*(d*x+c)^(1/2)*B*c*d^3+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+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((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*A*d^4+93/64/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(1/2)*A*d^4-3/8/(
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+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+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+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*d*x+a*d)^4/(a*d-b*c)/b*(d*x+c
)^(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((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*C*a*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((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*D*a^2*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^2-55/192/(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^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^3-5/64/(b*d*x+a*d)^4/(a*d-b*c)/b*(d*x+c)^(1/2)*a*B*d^4-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*b^3*c^3-5/64/(b*d*x+a*d)^4/(a*d-b*c)/b^3*(d*x+c)^(
1/2)*a^3*d^4*D-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/
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*b^3*c^2*d^2+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^4-55/24/(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^3-5/8/(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*b^3*c*d^3+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((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*a*B*d^4+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((d*x+c)^(1/2)/((a*
d-b*c)*b)^(1/2)*b)*a^3*d^4*D-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^2-3*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((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*a^2*
C*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^4-5/12/(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+3/64/(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+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^4-73/24/(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^3-11/64/(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+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+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^2-3*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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c positive or negative?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^5\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^5*(c + d*x)^(1/2)),x)

[Out]

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^5*(c + d*x)^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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