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

[Out]

-1/8*(a^3*d^3*D+a^2*b*d^2*(C*d-6*D*c)-a*b^2*d*(-5*B*d^2+12*C*c*d-24*D*c^2)-b^3*(35*A*d^3-30*B*c*d^2+24*C*c^2*d
-16*D*c^3))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(5/2)/(-a*d+b*c)^(9/2)+1/3*(a*b^2*B*d^3-a^2*b*C*
d^3+a^3*d^3*D-b^3*(7*A*d^3-6*B*c*d^2+6*C*c^2*d-6*D*c^3))/b^3/(-a*d+b*c)^4/(d*x+c)^(1/2)+1/3*(-A*b^3+a*(B*b^2-C
*a*b+D*a^2))/b^3/(-a*d+b*c)/(b*x+a)^3/(d*x+c)^(1/2)-1/12*(b^3*(-7*A*d+6*B*c)-a*b^2*(-B*d+12*C*c)-11*a^3*d*D+a^
2*b*(5*C*d+18*D*c))*(d*x+c)^(1/2)/b^2/(-a*d+b*c)^3/(b*x+a)^2-1/24*(b^3*(49*A*d^2-42*B*c*d+24*C*c^2)+5*a^3*d^2*
D-a^2*b*d*(11*C*d-18*D*c)+a*b^2*(-7*B*d^2+36*C*c*d-72*D*c^2))*(d*x+c)^(1/2)/b^2/(-a*d+b*c)^4/(b*x+a)

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Rubi [A]  time = 1.15, antiderivative size = 463, 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, 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 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 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 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 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 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)^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*}

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Mathematica [A]  time = 2.34, size = 697, normalized size = 1.51 \[ \frac {\sqrt {c+d x} \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{3 b^2 (a+b x)^3 (b c-a d)^2}+\frac {5 d \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \left (3 d^2 (a+b x)^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )+\sqrt {b} \sqrt {c+d x} \sqrt {b c-a d} (-5 a d+2 b c-3 b d x)\right )}{24 b^{5/2} (a+b x)^2 (b c-a d)^{9/2}}+\frac {\sqrt {c+d x} \left (a^3 d^2 D-3 a^2 b c d D+3 a b^2 c^2 D-b^3 \left (A d^2-B c d+c^2 C\right )\right )}{b^2 (a+b x) (b c-a d)^4}-\frac {\sqrt {c+d x} \left (-2 a^3 d D+a^2 b (3 c D+C d)-2 a b^2 c C+b^3 (B c-A d)\right )}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (a^3 \left (-d^2\right ) D+3 a^2 b c d D-3 a b^2 c^2 D+b^3 \left (A d^2-B c d+c^2 C\right )\right )}{b^{5/2} (b c-a d)^{9/2}}+\frac {3 d \left (d (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )-\sqrt {b} \sqrt {c+d x} \sqrt {b c-a d}\right ) \left (2 a^3 d D-a^2 b (3 c D+C d)+2 a b^2 c C+b^3 (A d-B c)\right )}{4 b^{5/2} (a+b x) (b c-a d)^{9/2}}+\frac {2 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )}{\sqrt {c+d x} (b c-a d)^4}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{(b c-a d)^{9/2}} \]

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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fricas [B]  time = 1.25, size = 3834, normalized size = 8.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*(16*D*a^3*b^3*c^4 + (16*D*b^6*c^3*d + (D*a^3*b^3 + C*a^2*b^4 + 5*B*a*b^5 - 35*A*b^6)*d^4 - 6*(D*a^2*b
^4*c + (2*C*a*b^5 - 5*B*b^6)*c)*d^3 + 24*(D*a*b^5*c^2 - C*b^6*c^2)*d^2)*x^4 + (D*a^6*c + (C*a^5*b + 5*B*a^4*b^
2 - 35*A*a^3*b^3)*c)*d^3 + (16*D*b^6*c^4 + 3*(D*a^4*b^2 + C*a^3*b^3 + 5*B*a^2*b^4 - 35*A*a*b^5)*d^4 - (17*D*a^
3*b^3*c + 5*(7*C*a^2*b^4 - 19*B*a*b^5 + 7*A*b^6)*c)*d^3 + 6*(11*D*a^2*b^4*c^2 - (14*C*a*b^5 - 5*B*b^6)*c^2)*d^
2 + 24*(3*D*a*b^5*c^3 - C*b^6*c^3)*d)*x^3 - 6*(D*a^5*b*c^2 + (2*C*a^4*b^2 - 5*B*a^3*b^3)*c^2)*d^2 + 3*(16*D*a*
b^5*c^4 + (D*a^5*b + C*a^4*b^2 + 5*B*a^3*b^3 - 35*A*a^2*b^4)*d^4 - (5*D*a^4*b^2*c + (11*C*a^3*b^3 - 35*B*a^2*b
^4 + 35*A*a*b^5)*c)*d^3 + 6*(3*D*a^3*b^3*c^2 - (6*C*a^2*b^4 - 5*B*a*b^5)*c^2)*d^2 + 8*(5*D*a^2*b^4*c^3 - 3*C*a
*b^5*c^3)*d)*x^2 + 24*(D*a^4*b^2*c^3 - C*a^3*b^3*c^3)*d + (48*D*a^2*b^4*c^4 + (D*a^6 + C*a^5*b + 5*B*a^4*b^2 -
35*A*a^3*b^3)*d^4 - 3*(D*a^5*b*c + (3*C*a^4*b^2 - 15*B*a^3*b^3 + 35*A*a^2*b^4)*c)*d^3 + 6*(D*a^4*b^2*c^2 - 5*
(2*C*a^3*b^3 - 3*B*a^2*b^4)*c^2)*d^2 + 8*(11*D*a^3*b^3*c^3 - 9*C*a^2*b^4*c^3)*d)*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*(92*D*a^3*b^4*c^4 + 48*A*a^4*b^3*d^4
- 4*(2*C*a^2*b^5 + B*a*b^6 + 2*A*b^7)*c^4 + 3*(D*a^6*b*c + (C*a^5*b^2 - 27*B*a^4*b^3 + 13*A*a^3*b^4)*c)*d^3 +
3*(16*D*b^7*c^4 - (D*a^4*b^3 + C*a^3*b^4 + 5*B*a^2*b^5 - 35*A*a*b^6)*d^4 + (7*D*a^3*b^4*c + (13*C*a^2*b^5 - 25
*B*a*b^6 - 35*A*b^7)*c)*d^3 - 6*(5*D*a^2*b^5*c^2 - (2*C*a*b^6 + 5*B*b^7)*c^2)*d^2 + 8*(D*a*b^6*c^3 - 3*C*b^7*c
^3)*d)*x^3 - (19*D*a^5*b^2*c^2 - (91*C*a^4*b^3 + 53*B*a^3*b^4 - 125*A*a^2*b^5)*c^2)*d^2 + (216*D*a*b^6*c^4 - 2
4*C*b^7*c^4 + 8*(D*a^5*b^2 - C*a^4*b^3 - 5*B*a^3*b^4 + 35*A*a^2*b^5)*d^4 - (25*D*a^4*b^3*c - (103*C*a^3*b^4 -
205*B*a^2*b^5 - 245*A*a*b^6)*c)*d^3 - (73*D*a^3*b^4*c^2 - (109*C*a^2*b^5 + 215*B*a*b^6 - 35*A*b^7)*c^2)*d^2 -
6*(21*D*a^2*b^5*c^3 + 5*(6*C*a*b^6 - B*b^7)*c^3)*d)*x^2 - 2*(38*D*a^4*b^3*c^3 + (43*C*a^3*b^4 - 16*B*a^2*b^5 -
23*A*a*b^6)*c^3)*d + (252*D*a^2*b^5*c^4 - 12*(2*C*a*b^6 + B*b^7)*c^4 + 3*(D*a^6*b + C*a^5*b^2 - 11*B*a^4*b^3
+ 77*A*a^3*b^4)*d^4 - (11*D*a^5*b^2*c - (35*C*a^4*b^3 - 179*B*a^3*b^4 - 133*A*a^2*b^5)*c)*d^3 - 2*(25*D*a^4*b^
3*c^2 - (106*C*a^3*b^4 + 65*B*a^2*b^5 - 56*A*a*b^6)*c^2)*d^2 - 2*(97*D*a^3*b^4*c^3 + (113*C*a^2*b^5 - 47*B*a*b
^6 - 7*A*b^7)*c^3)*d)*x)*sqrt(d*x + c))/(a^3*b^8*c^6 - 5*a^4*b^7*c^5*d + 10*a^5*b^6*c^4*d^2 - 10*a^6*b^5*c^3*d
^3 + 5*a^7*b^4*c^2*d^4 - a^8*b^3*c*d^5 + (b^11*c^5*d - 5*a*b^10*c^4*d^2 + 10*a^2*b^9*c^3*d^3 - 10*a^3*b^8*c^2*
d^4 + 5*a^4*b^7*c*d^5 - a^5*b^6*d^6)*x^4 + (b^11*c^6 - 2*a*b^10*c^5*d - 5*a^2*b^9*c^4*d^2 + 20*a^3*b^8*c^3*d^3
- 25*a^4*b^7*c^2*d^4 + 14*a^5*b^6*c*d^5 - 3*a^6*b^5*d^6)*x^3 + 3*(a*b^10*c^6 - 4*a^2*b^9*c^5*d + 5*a^3*b^8*c^
4*d^2 - 5*a^5*b^6*c^2*d^4 + 4*a^6*b^5*c*d^5 - a^7*b^4*d^6)*x^2 + (3*a^2*b^9*c^6 - 14*a^3*b^8*c^5*d + 25*a^4*b^
7*c^4*d^2 - 20*a^5*b^6*c^3*d^3 + 5*a^6*b^5*c^2*d^4 + 2*a^7*b^4*c*d^5 - a^8*b^3*d^6)*x), 1/24*(3*(16*D*a^3*b^3*
c^4 + (16*D*b^6*c^3*d + (D*a^3*b^3 + C*a^2*b^4 + 5*B*a*b^5 - 35*A*b^6)*d^4 - 6*(D*a^2*b^4*c + (2*C*a*b^5 - 5*B
*b^6)*c)*d^3 + 24*(D*a*b^5*c^2 - C*b^6*c^2)*d^2)*x^4 + (D*a^6*c + (C*a^5*b + 5*B*a^4*b^2 - 35*A*a^3*b^3)*c)*d^
3 + (16*D*b^6*c^4 + 3*(D*a^4*b^2 + C*a^3*b^3 + 5*B*a^2*b^4 - 35*A*a*b^5)*d^4 - (17*D*a^3*b^3*c + 5*(7*C*a^2*b^
4 - 19*B*a*b^5 + 7*A*b^6)*c)*d^3 + 6*(11*D*a^2*b^4*c^2 - (14*C*a*b^5 - 5*B*b^6)*c^2)*d^2 + 24*(3*D*a*b^5*c^3 -
C*b^6*c^3)*d)*x^3 - 6*(D*a^5*b*c^2 + (2*C*a^4*b^2 - 5*B*a^3*b^3)*c^2)*d^2 + 3*(16*D*a*b^5*c^4 + (D*a^5*b + C*
a^4*b^2 + 5*B*a^3*b^3 - 35*A*a^2*b^4)*d^4 - (5*D*a^4*b^2*c + (11*C*a^3*b^3 - 35*B*a^2*b^4 + 35*A*a*b^5)*c)*d^3
+ 6*(3*D*a^3*b^3*c^2 - (6*C*a^2*b^4 - 5*B*a*b^5)*c^2)*d^2 + 8*(5*D*a^2*b^4*c^3 - 3*C*a*b^5*c^3)*d)*x^2 + 24*(
D*a^4*b^2*c^3 - C*a^3*b^3*c^3)*d + (48*D*a^2*b^4*c^4 + (D*a^6 + C*a^5*b + 5*B*a^4*b^2 - 35*A*a^3*b^3)*d^4 - 3*
(D*a^5*b*c + (3*C*a^4*b^2 - 15*B*a^3*b^3 + 35*A*a^2*b^4)*c)*d^3 + 6*(D*a^4*b^2*c^2 - 5*(2*C*a^3*b^3 - 3*B*a^2*
b^4)*c^2)*d^2 + 8*(11*D*a^3*b^3*c^3 - 9*C*a^2*b^4*c^3)*d)*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)) + (92*D*a^3*b^4*c^4 + 48*A*a^4*b^3*d^4 - 4*(2*C*a^2*b^5 + B*a*b^6 + 2*A*b^7)*c^4
+ 3*(D*a^6*b*c + (C*a^5*b^2 - 27*B*a^4*b^3 + 13*A*a^3*b^4)*c)*d^3 + 3*(16*D*b^7*c^4 - (D*a^4*b^3 + C*a^3*b^4 +
5*B*a^2*b^5 - 35*A*a*b^6)*d^4 + (7*D*a^3*b^4*c + (13*C*a^2*b^5 - 25*B*a*b^6 - 35*A*b^7)*c)*d^3 - 6*(5*D*a^2*b
^5*c^2 - (2*C*a*b^6 + 5*B*b^7)*c^2)*d^2 + 8*(D*a*b^6*c^3 - 3*C*b^7*c^3)*d)*x^3 - (19*D*a^5*b^2*c^2 - (91*C*a^4
*b^3 + 53*B*a^3*b^4 - 125*A*a^2*b^5)*c^2)*d^2 + (216*D*a*b^6*c^4 - 24*C*b^7*c^4 + 8*(D*a^5*b^2 - C*a^4*b^3 - 5
*B*a^3*b^4 + 35*A*a^2*b^5)*d^4 - (25*D*a^4*b^3*c - (103*C*a^3*b^4 - 205*B*a^2*b^5 - 245*A*a*b^6)*c)*d^3 - (73*
D*a^3*b^4*c^2 - (109*C*a^2*b^5 + 215*B*a*b^6 - 35*A*b^7)*c^2)*d^2 - 6*(21*D*a^2*b^5*c^3 + 5*(6*C*a*b^6 - B*b^7
)*c^3)*d)*x^2 - 2*(38*D*a^4*b^3*c^3 + (43*C*a^3*b^4 - 16*B*a^2*b^5 - 23*A*a*b^6)*c^3)*d + (252*D*a^2*b^5*c^4 -
12*(2*C*a*b^6 + B*b^7)*c^4 + 3*(D*a^6*b + C*a^5*b^2 - 11*B*a^4*b^3 + 77*A*a^3*b^4)*d^4 - (11*D*a^5*b^2*c - (3
5*C*a^4*b^3 - 179*B*a^3*b^4 - 133*A*a^2*b^5)*c)*d^3 - 2*(25*D*a^4*b^3*c^2 - (106*C*a^3*b^4 + 65*B*a^2*b^5 - 56
*A*a*b^6)*c^2)*d^2 - 2*(97*D*a^3*b^4*c^3 + (113*C*a^2*b^5 - 47*B*a*b^6 - 7*A*b^7)*c^3)*d)*x)*sqrt(d*x + c))/(a
^3*b^8*c^6 - 5*a^4*b^7*c^5*d + 10*a^5*b^6*c^4*d^2 - 10*a^6*b^5*c^3*d^3 + 5*a^7*b^4*c^2*d^4 - a^8*b^3*c*d^5 + (
b^11*c^5*d - 5*a*b^10*c^4*d^2 + 10*a^2*b^9*c^3*d^3 - 10*a^3*b^8*c^2*d^4 + 5*a^4*b^7*c*d^5 - a^5*b^6*d^6)*x^4 +
(b^11*c^6 - 2*a*b^10*c^5*d - 5*a^2*b^9*c^4*d^2 + 20*a^3*b^8*c^3*d^3 - 25*a^4*b^7*c^2*d^4 + 14*a^5*b^6*c*d^5 -
3*a^6*b^5*d^6)*x^3 + 3*(a*b^10*c^6 - 4*a^2*b^9*c^5*d + 5*a^3*b^8*c^4*d^2 - 5*a^5*b^6*c^2*d^4 + 4*a^6*b^5*c*d^
5 - a^7*b^4*d^6)*x^2 + (3*a^2*b^9*c^6 - 14*a^3*b^8*c^5*d + 25*a^4*b^7*c^4*d^2 - 20*a^5*b^6*c^3*d^3 + 5*a^6*b^5
*c^2*d^4 + 2*a^7*b^4*c*d^5 - a^8*b^3*d^6)*x)]

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giac [B]  time = 1.56, size = 1085, normalized size = 2.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^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)

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maple [B]  time = 0.04, size = 2108, normalized size = 4.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^4/(d*x+c)^(3/2),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 \[ \int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^4\,{\left (c+d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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