3.17.50 \(\int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx\)

Optimal. Leaf size=260 \[ -\frac {2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3 (e+f x)^{3/2} (d e-c f)^3}+\frac {2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{5 f^3 (e+f x)^{5/2} (d e-c f)^2}-\frac {2 (b e-a f)^3}{7 f^3 (e+f x)^{7/2} (d e-c f)}-\frac {2 (b c-a d)^3}{\sqrt {e+f x} (d e-c f)^4}+\frac {2 \sqrt {d} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}} \]

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Rubi [A]  time = 0.36, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {87, 63, 208} \begin {gather*} -\frac {2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3 (e+f x)^{3/2} (d e-c f)^3}+\frac {2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{5 f^3 (e+f x)^{5/2} (d e-c f)^2}-\frac {2 (b e-a f)^3}{7 f^3 (e+f x)^{7/2} (d e-c f)}-\frac {2 (b c-a d)^3}{\sqrt {e+f x} (d e-c f)^4}+\frac {2 \sqrt {d} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/((c + d*x)*(e + f*x)^(9/2)),x]

[Out]

(-2*(b*e - a*f)^3)/(7*f^3*(d*e - c*f)*(e + f*x)^(7/2)) + (2*(b*e - a*f)^2*(2*b*d*e - 3*b*c*f + a*d*f))/(5*f^3*
(d*e - c*f)^2*(e + f*x)^(5/2)) - (2*(b*e - a*f)*(a^2*d^2*f^2 + a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*
f + 3*c^2*f^2)))/(3*f^3*(d*e - c*f)^3*(e + f*x)^(3/2)) - (2*(b*c - a*d)^3)/((d*e - c*f)^4*Sqrt[e + f*x]) + (2*
Sqrt[d]*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(9/2)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 87

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[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)^3}{(c+d x) (e+f x)^{9/2}} \, dx &=\int \left (\frac {(-b e+a f)^3}{f^2 (-d e+c f) (e+f x)^{9/2}}+\frac {(-b e+a f)^2 (-2 b d e+3 b c f-a d f)}{f^2 (-d e+c f)^2 (e+f x)^{7/2}}+\frac {-3 a b^2 c^2 f^3+3 a^2 b c d f^3-a^3 d^2 f^3+b^3 e \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )}{f^2 (d e-c f)^3 (e+f x)^{5/2}}+\frac {(b c-a d)^3 f}{(-d e+c f)^4 (e+f x)^{3/2}}+\frac {d (-b c+a d)^3}{(d e-c f)^4 (c+d x) \sqrt {e+f x}}\right ) \, dx\\ &=-\frac {2 (b e-a f)^3}{7 f^3 (d e-c f) (e+f x)^{7/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{5 f^3 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right )}{3 f^3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 (b c-a d)^3}{(d e-c f)^4 \sqrt {e+f x}}-\frac {\left (d (b c-a d)^3\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{(d e-c f)^4}\\ &=-\frac {2 (b e-a f)^3}{7 f^3 (d e-c f) (e+f x)^{7/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{5 f^3 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right )}{3 f^3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 (b c-a d)^3}{(d e-c f)^4 \sqrt {e+f x}}-\frac {\left (2 d (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{f (d e-c f)^4}\\ &=-\frac {2 (b e-a f)^3}{7 f^3 (d e-c f) (e+f x)^{7/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{5 f^3 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right )}{3 f^3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 (b c-a d)^3}{(d e-c f)^4 \sqrt {e+f x}}+\frac {2 \sqrt {d} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.19, size = 166, normalized size = 0.64 \begin {gather*} \frac {2 \left (-\frac {15 b \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{f^3}+\frac {21 b^2 d (e+f x) (-3 a d f+b c f+2 b d e)}{f^3}+\frac {15 (b c-a d)^3 \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\frac {d (e+f x)}{d e-c f}\right )}{c f-d e}-\frac {35 b^3 d^2 (e+f x)^2}{f^3}\right )}{105 d^3 (e+f x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/((c + d*x)*(e + f*x)^(9/2)),x]

[Out]

(2*((-15*b*(3*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*d*e*f + c^2*f^2)))/f^3 + (21*b^2*d*(2*b*d
*e + b*c*f - 3*a*d*f)*(e + f*x))/f^3 - (35*b^3*d^2*(e + f*x)^2)/f^3 + (15*(b*c - a*d)^3*Hypergeometric2F1[-7/2
, 1, -5/2, (d*(e + f*x))/(d*e - c*f)])/(-(d*e) + c*f)))/(105*d^3*(e + f*x)^(7/2))

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IntegrateAlgebraic [B]  time = 0.58, size = 857, normalized size = 3.30 \begin {gather*} -\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c f-d e} \sqrt {e+f x}}{d e-c f}\right ) (a d-b c)^3}{(c f-d e)^{9/2}}-\frac {2 \left (15 b^3 d^3 e^6-45 a b^2 d^3 f e^5-45 b^3 c d^2 f e^5-42 b^3 d^3 (e+f x) e^5+45 a^2 b d^3 f^2 e^4+135 a b^2 c d^2 f^2 e^4+45 b^3 c^2 d f^2 e^4+35 b^3 d^3 (e+f x)^2 e^4+63 a b^2 d^3 f (e+f x) e^4+147 b^3 c d^2 f (e+f x) e^4-15 b^3 c^3 f^3 e^3-15 a^3 d^3 f^3 e^3-135 a^2 b c d^2 f^3 e^3-135 a b^2 c^2 d f^3 e^3-140 b^3 c d^2 f (e+f x)^2 e^3-252 a b^2 c d^2 f^2 (e+f x) e^3-168 b^3 c^2 d f^2 (e+f x) e^3+45 a b^2 c^3 f^4 e^2+45 a^3 c d^2 f^4 e^2+135 a^2 b c^2 d f^4 e^2+210 b^3 c^2 d f^2 (e+f x)^2 e^2+63 b^3 c^3 f^3 (e+f x) e^2-21 a^3 d^3 f^3 (e+f x) e^2+63 a^2 b c d^2 f^3 (e+f x) e^2+315 a b^2 c^2 d f^3 (e+f x) e^2-45 a^2 b c^3 f^5 e-45 a^3 c^2 d f^5 e-105 b^3 c^3 f^3 (e+f x)^2 e-35 a^3 d^3 f^3 (e+f x)^2 e+105 a^2 b c d^2 f^3 (e+f x)^2 e-105 a b^2 c^2 d f^3 (e+f x)^2 e-126 a b^2 c^3 f^4 (e+f x) e+42 a^3 c d^2 f^4 (e+f x) e-126 a^2 b c^2 d f^4 (e+f x) e+15 a^3 c^3 f^6+105 b^3 c^3 f^3 (e+f x)^3-105 a^3 d^3 f^3 (e+f x)^3+315 a^2 b c d^2 f^3 (e+f x)^3-315 a b^2 c^2 d f^3 (e+f x)^3+105 a b^2 c^3 f^4 (e+f x)^2+35 a^3 c d^2 f^4 (e+f x)^2-105 a^2 b c^2 d f^4 (e+f x)^2+63 a^2 b c^3 f^5 (e+f x)-21 a^3 c^2 d f^5 (e+f x)\right )}{105 f^3 (c f-d e)^4 (e+f x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^3/((c + d*x)*(e + f*x)^(9/2)),x]

[Out]

(-2*(15*b^3*d^3*e^6 - 45*b^3*c*d^2*e^5*f - 45*a*b^2*d^3*e^5*f + 45*b^3*c^2*d*e^4*f^2 + 135*a*b^2*c*d^2*e^4*f^2
+ 45*a^2*b*d^3*e^4*f^2 - 15*b^3*c^3*e^3*f^3 - 135*a*b^2*c^2*d*e^3*f^3 - 135*a^2*b*c*d^2*e^3*f^3 - 15*a^3*d^3*
e^3*f^3 + 45*a*b^2*c^3*e^2*f^4 + 135*a^2*b*c^2*d*e^2*f^4 + 45*a^3*c*d^2*e^2*f^4 - 45*a^2*b*c^3*e*f^5 - 45*a^3*
c^2*d*e*f^5 + 15*a^3*c^3*f^6 - 42*b^3*d^3*e^5*(e + f*x) + 147*b^3*c*d^2*e^4*f*(e + f*x) + 63*a*b^2*d^3*e^4*f*(
e + f*x) - 168*b^3*c^2*d*e^3*f^2*(e + f*x) - 252*a*b^2*c*d^2*e^3*f^2*(e + f*x) + 63*b^3*c^3*e^2*f^3*(e + f*x)
+ 315*a*b^2*c^2*d*e^2*f^3*(e + f*x) + 63*a^2*b*c*d^2*e^2*f^3*(e + f*x) - 21*a^3*d^3*e^2*f^3*(e + f*x) - 126*a*
b^2*c^3*e*f^4*(e + f*x) - 126*a^2*b*c^2*d*e*f^4*(e + f*x) + 42*a^3*c*d^2*e*f^4*(e + f*x) + 63*a^2*b*c^3*f^5*(e
+ f*x) - 21*a^3*c^2*d*f^5*(e + f*x) + 35*b^3*d^3*e^4*(e + f*x)^2 - 140*b^3*c*d^2*e^3*f*(e + f*x)^2 + 210*b^3*
c^2*d*e^2*f^2*(e + f*x)^2 - 105*b^3*c^3*e*f^3*(e + f*x)^2 - 105*a*b^2*c^2*d*e*f^3*(e + f*x)^2 + 105*a^2*b*c*d^
2*e*f^3*(e + f*x)^2 - 35*a^3*d^3*e*f^3*(e + f*x)^2 + 105*a*b^2*c^3*f^4*(e + f*x)^2 - 105*a^2*b*c^2*d*f^4*(e +
f*x)^2 + 35*a^3*c*d^2*f^4*(e + f*x)^2 + 105*b^3*c^3*f^3*(e + f*x)^3 - 315*a*b^2*c^2*d*f^3*(e + f*x)^3 + 315*a^
2*b*c*d^2*f^3*(e + f*x)^3 - 105*a^3*d^3*f^3*(e + f*x)^3))/(105*f^3*(-(d*e) + c*f)^4*(e + f*x)^(7/2)) - (2*Sqrt
[d]*(-(b*c) + a*d)^3*ArcTan[(Sqrt[d]*Sqrt[-(d*e) + c*f]*Sqrt[e + f*x])/(d*e - c*f)])/(-(d*e) + c*f)^(9/2)

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fricas [B]  time = 1.90, size = 2284, normalized size = 8.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x, algorithm="fricas")

[Out]

[-1/105*(105*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^7*x^4 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2
*b*c*d^2 - a^3*d^3)*e*f^6*x^3 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^5*x^2 + 4*(b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^3*f^4*x + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^
4*f^3)*sqrt(d/(d*e - c*f))*log((d*f*x + 2*d*e - c*f - 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x +
c)) + 2*(8*b^3*d^3*e^6 + 15*a^3*c^3*f^6 + 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^6*x^3 - 2*
(19*b^3*c*d^2 - 9*a*b^2*d^3)*e^5*f + 3*(29*b^3*c^2*d - 39*a*b^2*c*d^2 + 15*a^2*b*d^3)*e^4*f^2 + 4*(12*b^3*c^3
- 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 44*a^3*d^3)*e^3*f^3 + 2*(12*a*b^2*c^3 - 48*a^2*b*c^2*d + 61*a^3*c*d^2)*e^2
*f^4 + 6*(3*a^2*b*c^3 - 11*a^3*c^2*d)*e*f^5 + 35*(b^3*d^3*e^4*f^2 - 4*b^3*c*d^2*e^3*f^3 + 6*b^3*c^2*d*e^2*f^4
+ 2*(3*b^3*c^3 - 15*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 5*a^3*d^3)*e*f^5 + (3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2
)*f^6)*x^2 + 7*(4*b^3*d^3*e^5*f - (19*b^3*c*d^2 - 9*a*b^2*d^3)*e^4*f^2 + 36*(b^3*c^2*d - a*b^2*c*d^2)*e^3*f^3
+ 2*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 29*a^3*d^3)*e^2*f^4 + 4*(3*a*b^2*c^3 - 12*a^2*b*c^2*d + 4*
a^3*c*d^2)*e*f^5 + 3*(3*a^2*b*c^3 - a^3*c^2*d)*f^6)*x)*sqrt(f*x + e))/(d^4*e^8*f^3 - 4*c*d^3*e^7*f^4 + 6*c^2*d
^2*e^6*f^5 - 4*c^3*d*e^5*f^6 + c^4*e^4*f^7 + (d^4*e^4*f^7 - 4*c*d^3*e^3*f^8 + 6*c^2*d^2*e^2*f^9 - 4*c^3*d*e*f^
10 + c^4*f^11)*x^4 + 4*(d^4*e^5*f^6 - 4*c*d^3*e^4*f^7 + 6*c^2*d^2*e^3*f^8 - 4*c^3*d*e^2*f^9 + c^4*e*f^10)*x^3
+ 6*(d^4*e^6*f^5 - 4*c*d^3*e^5*f^6 + 6*c^2*d^2*e^4*f^7 - 4*c^3*d*e^3*f^8 + c^4*e^2*f^9)*x^2 + 4*(d^4*e^7*f^4 -
4*c*d^3*e^6*f^5 + 6*c^2*d^2*e^5*f^6 - 4*c^3*d*e^4*f^7 + c^4*e^3*f^8)*x), 2/105*(105*((b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*f^7*x^4 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^6*x^3 + 6*(b^3
*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^5*x^2 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3
*d^3)*e^3*f^4*x + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^4*f^3)*sqrt(-d/(d*e - c*f))*arctan(-(d
*e - c*f)*sqrt(f*x + e)*sqrt(-d/(d*e - c*f))/(d*f*x + d*e)) - (8*b^3*d^3*e^6 + 15*a^3*c^3*f^6 + 105*(b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^6*x^3 - 2*(19*b^3*c*d^2 - 9*a*b^2*d^3)*e^5*f + 3*(29*b^3*c^2*d - 3
9*a*b^2*c*d^2 + 15*a^2*b*d^3)*e^4*f^2 + 4*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 44*a^3*d^3)*e^3*f^3
+ 2*(12*a*b^2*c^3 - 48*a^2*b*c^2*d + 61*a^3*c*d^2)*e^2*f^4 + 6*(3*a^2*b*c^3 - 11*a^3*c^2*d)*e*f^5 + 35*(b^3*d^
3*e^4*f^2 - 4*b^3*c*d^2*e^3*f^3 + 6*b^3*c^2*d*e^2*f^4 + 2*(3*b^3*c^3 - 15*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 5*a^3
*d^3)*e*f^5 + (3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*f^6)*x^2 + 7*(4*b^3*d^3*e^5*f - (19*b^3*c*d^2 - 9*a*b^
2*d^3)*e^4*f^2 + 36*(b^3*c^2*d - a*b^2*c*d^2)*e^3*f^3 + 2*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 29*a
^3*d^3)*e^2*f^4 + 4*(3*a*b^2*c^3 - 12*a^2*b*c^2*d + 4*a^3*c*d^2)*e*f^5 + 3*(3*a^2*b*c^3 - a^3*c^2*d)*f^6)*x)*s
qrt(f*x + e))/(d^4*e^8*f^3 - 4*c*d^3*e^7*f^4 + 6*c^2*d^2*e^6*f^5 - 4*c^3*d*e^5*f^6 + c^4*e^4*f^7 + (d^4*e^4*f^
7 - 4*c*d^3*e^3*f^8 + 6*c^2*d^2*e^2*f^9 - 4*c^3*d*e*f^10 + c^4*f^11)*x^4 + 4*(d^4*e^5*f^6 - 4*c*d^3*e^4*f^7 +
6*c^2*d^2*e^3*f^8 - 4*c^3*d*e^2*f^9 + c^4*e*f^10)*x^3 + 6*(d^4*e^6*f^5 - 4*c*d^3*e^5*f^6 + 6*c^2*d^2*e^4*f^7 -
4*c^3*d*e^3*f^8 + c^4*e^2*f^9)*x^2 + 4*(d^4*e^7*f^4 - 4*c*d^3*e^6*f^5 + 6*c^2*d^2*e^5*f^6 - 4*c^3*d*e^4*f^7 +
c^4*e^3*f^8)*x)]

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giac [B]  time = 1.37, size = 969, normalized size = 3.73 \begin {gather*} -\frac {2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{{\left (c^{4} f^{4} - 4 \, c^{3} d f^{3} e + 6 \, c^{2} d^{2} f^{2} e^{2} - 4 \, c d^{3} f e^{3} + d^{4} e^{4}\right )} \sqrt {c d f - d^{2} e}} - \frac {2 \, {\left (105 \, {\left (f x + e\right )}^{3} b^{3} c^{3} f^{3} - 315 \, {\left (f x + e\right )}^{3} a b^{2} c^{2} d f^{3} + 315 \, {\left (f x + e\right )}^{3} a^{2} b c d^{2} f^{3} - 105 \, {\left (f x + e\right )}^{3} a^{3} d^{3} f^{3} + 105 \, {\left (f x + e\right )}^{2} a b^{2} c^{3} f^{4} - 105 \, {\left (f x + e\right )}^{2} a^{2} b c^{2} d f^{4} + 35 \, {\left (f x + e\right )}^{2} a^{3} c d^{2} f^{4} + 63 \, {\left (f x + e\right )} a^{2} b c^{3} f^{5} - 21 \, {\left (f x + e\right )} a^{3} c^{2} d f^{5} + 15 \, a^{3} c^{3} f^{6} - 105 \, {\left (f x + e\right )}^{2} b^{3} c^{3} f^{3} e - 105 \, {\left (f x + e\right )}^{2} a b^{2} c^{2} d f^{3} e + 105 \, {\left (f x + e\right )}^{2} a^{2} b c d^{2} f^{3} e - 35 \, {\left (f x + e\right )}^{2} a^{3} d^{3} f^{3} e - 126 \, {\left (f x + e\right )} a b^{2} c^{3} f^{4} e - 126 \, {\left (f x + e\right )} a^{2} b c^{2} d f^{4} e + 42 \, {\left (f x + e\right )} a^{3} c d^{2} f^{4} e - 45 \, a^{2} b c^{3} f^{5} e - 45 \, a^{3} c^{2} d f^{5} e + 210 \, {\left (f x + e\right )}^{2} b^{3} c^{2} d f^{2} e^{2} + 63 \, {\left (f x + e\right )} b^{3} c^{3} f^{3} e^{2} + 315 \, {\left (f x + e\right )} a b^{2} c^{2} d f^{3} e^{2} + 63 \, {\left (f x + e\right )} a^{2} b c d^{2} f^{3} e^{2} - 21 \, {\left (f x + e\right )} a^{3} d^{3} f^{3} e^{2} + 45 \, a b^{2} c^{3} f^{4} e^{2} + 135 \, a^{2} b c^{2} d f^{4} e^{2} + 45 \, a^{3} c d^{2} f^{4} e^{2} - 140 \, {\left (f x + e\right )}^{2} b^{3} c d^{2} f e^{3} - 168 \, {\left (f x + e\right )} b^{3} c^{2} d f^{2} e^{3} - 252 \, {\left (f x + e\right )} a b^{2} c d^{2} f^{2} e^{3} - 15 \, b^{3} c^{3} f^{3} e^{3} - 135 \, a b^{2} c^{2} d f^{3} e^{3} - 135 \, a^{2} b c d^{2} f^{3} e^{3} - 15 \, a^{3} d^{3} f^{3} e^{3} + 35 \, {\left (f x + e\right )}^{2} b^{3} d^{3} e^{4} + 147 \, {\left (f x + e\right )} b^{3} c d^{2} f e^{4} + 63 \, {\left (f x + e\right )} a b^{2} d^{3} f e^{4} + 45 \, b^{3} c^{2} d f^{2} e^{4} + 135 \, a b^{2} c d^{2} f^{2} e^{4} + 45 \, a^{2} b d^{3} f^{2} e^{4} - 42 \, {\left (f x + e\right )} b^{3} d^{3} e^{5} - 45 \, b^{3} c d^{2} f e^{5} - 45 \, a b^{2} d^{3} f e^{5} + 15 \, b^{3} d^{3} e^{6}\right )}}{105 \, {\left (c^{4} f^{7} - 4 \, c^{3} d f^{6} e + 6 \, c^{2} d^{2} f^{5} e^{2} - 4 \, c d^{3} f^{4} e^{3} + d^{4} f^{3} e^{4}\right )} {\left (f x + e\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x, algorithm="giac")

[Out]

-2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/((c^4*f
^4 - 4*c^3*d*f^3*e + 6*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 + d^4*e^4)*sqrt(c*d*f - d^2*e)) - 2/105*(105*(f*x + e)^
3*b^3*c^3*f^3 - 315*(f*x + e)^3*a*b^2*c^2*d*f^3 + 315*(f*x + e)^3*a^2*b*c*d^2*f^3 - 105*(f*x + e)^3*a^3*d^3*f^
3 + 105*(f*x + e)^2*a*b^2*c^3*f^4 - 105*(f*x + e)^2*a^2*b*c^2*d*f^4 + 35*(f*x + e)^2*a^3*c*d^2*f^4 + 63*(f*x +
e)*a^2*b*c^3*f^5 - 21*(f*x + e)*a^3*c^2*d*f^5 + 15*a^3*c^3*f^6 - 105*(f*x + e)^2*b^3*c^3*f^3*e - 105*(f*x + e
)^2*a*b^2*c^2*d*f^3*e + 105*(f*x + e)^2*a^2*b*c*d^2*f^3*e - 35*(f*x + e)^2*a^3*d^3*f^3*e - 126*(f*x + e)*a*b^2
*c^3*f^4*e - 126*(f*x + e)*a^2*b*c^2*d*f^4*e + 42*(f*x + e)*a^3*c*d^2*f^4*e - 45*a^2*b*c^3*f^5*e - 45*a^3*c^2*
d*f^5*e + 210*(f*x + e)^2*b^3*c^2*d*f^2*e^2 + 63*(f*x + e)*b^3*c^3*f^3*e^2 + 315*(f*x + e)*a*b^2*c^2*d*f^3*e^2
+ 63*(f*x + e)*a^2*b*c*d^2*f^3*e^2 - 21*(f*x + e)*a^3*d^3*f^3*e^2 + 45*a*b^2*c^3*f^4*e^2 + 135*a^2*b*c^2*d*f^
4*e^2 + 45*a^3*c*d^2*f^4*e^2 - 140*(f*x + e)^2*b^3*c*d^2*f*e^3 - 168*(f*x + e)*b^3*c^2*d*f^2*e^3 - 252*(f*x +
e)*a*b^2*c*d^2*f^2*e^3 - 15*b^3*c^3*f^3*e^3 - 135*a*b^2*c^2*d*f^3*e^3 - 135*a^2*b*c*d^2*f^3*e^3 - 15*a^3*d^3*f
^3*e^3 + 35*(f*x + e)^2*b^3*d^3*e^4 + 147*(f*x + e)*b^3*c*d^2*f*e^4 + 63*(f*x + e)*a*b^2*d^3*f*e^4 + 45*b^3*c^
2*d*f^2*e^4 + 135*a*b^2*c*d^2*f^2*e^4 + 45*a^2*b*d^3*f^2*e^4 - 42*(f*x + e)*b^3*d^3*e^5 - 45*b^3*c*d^2*f*e^5 -
45*a*b^2*d^3*f*e^5 + 15*b^3*d^3*e^6)/((c^4*f^7 - 4*c^3*d*f^6*e + 6*c^2*d^2*f^5*e^2 - 4*c*d^3*f^4*e^3 + d^4*f^
3*e^4)*(f*x + e)^(7/2))

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maple [B]  time = 0.03, size = 756, normalized size = 2.91 \begin {gather*} \frac {2 a^{3} d^{4} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {6 a^{2} b c \,d^{3} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}+\frac {6 a \,b^{2} c^{2} d^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 b^{3} c^{3} d \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}+\frac {2 a^{3} d^{3}}{\left (c f -d e \right )^{4} \sqrt {f x +e}}-\frac {6 a^{2} b c \,d^{2}}{\left (c f -d e \right )^{4} \sqrt {f x +e}}+\frac {6 a \,b^{2} c^{2} d}{\left (c f -d e \right )^{4} \sqrt {f x +e}}-\frac {2 b^{3} c^{3}}{\left (c f -d e \right )^{4} \sqrt {f x +e}}-\frac {2 a^{3} d^{2}}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 a^{2} b c d}{\left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}-\frac {2 a \,b^{2} c^{2}}{\left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 b^{3} c^{2} e}{\left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}} f}-\frac {2 b^{3} c d \,e^{2}}{\left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}} f^{2}}+\frac {2 b^{3} d^{2} e^{3}}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}} f^{3}}+\frac {2 a^{3} d}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}-\frac {6 a^{2} b c}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}+\frac {12 a \,b^{2} c e}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}} f}-\frac {6 a \,b^{2} d \,e^{2}}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}} f^{2}}-\frac {6 b^{3} c \,e^{2}}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}} f^{2}}+\frac {4 b^{3} d \,e^{3}}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}} f^{3}}-\frac {2 a^{3}}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}}}+\frac {6 a^{2} b e}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}} f}-\frac {6 a \,b^{2} e^{2}}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}} f^{2}}+\frac {2 b^{3} e^{3}}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}} f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x)

[Out]

-2/7/(c*f-d*e)/(f*x+e)^(7/2)*a^3+6/7/f/(c*f-d*e)/(f*x+e)^(7/2)*a^2*b*e-6/7/f^2/(c*f-d*e)/(f*x+e)^(7/2)*a*b^2*e
^2+2/7/f^3/(c*f-d*e)/(f*x+e)^(7/2)*b^3*e^3+2/5/(c*f-d*e)^2/(f*x+e)^(5/2)*a^3*d-6/5/(c*f-d*e)^2/(f*x+e)^(5/2)*a
^2*b*c+12/5/f/(c*f-d*e)^2/(f*x+e)^(5/2)*a*b^2*c*e-6/5/f^2/(c*f-d*e)^2/(f*x+e)^(5/2)*a*b^2*d*e^2-6/5/f^2/(c*f-d
*e)^2/(f*x+e)^(5/2)*b^3*c*e^2+4/5/f^3/(c*f-d*e)^2/(f*x+e)^(5/2)*b^3*d*e^3-2/3/(c*f-d*e)^3/(f*x+e)^(3/2)*a^3*d^
2+2/(c*f-d*e)^3/(f*x+e)^(3/2)*a^2*b*c*d-2/(c*f-d*e)^3/(f*x+e)^(3/2)*a*b^2*c^2+2/f/(c*f-d*e)^3/(f*x+e)^(3/2)*b^
3*c^2*e-2/f^2/(c*f-d*e)^3/(f*x+e)^(3/2)*b^3*c*d*e^2+2/3/f^3/(c*f-d*e)^3/(f*x+e)^(3/2)*b^3*d^2*e^3+2/(c*f-d*e)^
4/(f*x+e)^(1/2)*a^3*d^3-6/(c*f-d*e)^4/(f*x+e)^(1/2)*a^2*b*c*d^2+6/(c*f-d*e)^4/(f*x+e)^(1/2)*a*b^2*c^2*d-2/(c*f
-d*e)^4/(f*x+e)^(1/2)*b^3*c^3+2*d^4/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d
)*a^3-6*d^3/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^2*b*c+6*d^2/(c*f-d*e
)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a*b^2*c^2-2*d/(c*f-d*e)^4/((c*f-d*e)*d)^(1
/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*b^3*c^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),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(c*f-d*e>0)', see `assume?` for
more details)Is c*f-d*e positive or negative?

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mupad [B]  time = 1.57, size = 438, normalized size = 1.68 \begin {gather*} \frac {2\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\left (c^4\,f^4-4\,c^3\,d\,e\,f^3+6\,c^2\,d^2\,e^2\,f^2-4\,c\,d^3\,e^3\,f+d^4\,e^4\right )}{{\left (c\,f-d\,e\right )}^{9/2}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (c\,f-d\,e\right )}^{9/2}}-\frac {\frac {2\,\left (a^3\,f^3-3\,a^2\,b\,e\,f^2+3\,a\,b^2\,e^2\,f-b^3\,e^3\right )}{7\,\left (c\,f-d\,e\right )}-\frac {2\,{\left (e+f\,x\right )}^3\,\left (a^3\,d^3\,f^3-3\,a^2\,b\,c\,d^2\,f^3+3\,a\,b^2\,c^2\,d\,f^3-b^3\,c^3\,f^3\right )}{{\left (c\,f-d\,e\right )}^4}+\frac {2\,{\left (e+f\,x\right )}^2\,\left (a^3\,d^2\,f^3-3\,a^2\,b\,c\,d\,f^3+3\,a\,b^2\,c^2\,f^3-3\,b^3\,c^2\,e\,f^2+3\,b^3\,c\,d\,e^2\,f-b^3\,d^2\,e^3\right )}{3\,{\left (c\,f-d\,e\right )}^3}-\frac {2\,\left (e+f\,x\right )\,\left (d\,a^3\,f^3-3\,c\,a^2\,b\,f^3-3\,d\,a\,b^2\,e^2\,f+6\,c\,a\,b^2\,e\,f^2+2\,d\,b^3\,e^3-3\,c\,b^3\,e^2\,f\right )}{5\,{\left (c\,f-d\,e\right )}^2}}{f^3\,{\left (e+f\,x\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/((e + f*x)^(9/2)*(c + d*x)),x)

[Out]

(2*d^(1/2)*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^3*(c^4*f^4 + d^4*e^4 + 6*c^2*d^2*e^2*f^2 - 4*c*d^3*e^3*f
- 4*c^3*d*e*f^3))/((c*f - d*e)^(9/2)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))*(a*d - b*c)^3)/(c*f
- d*e)^(9/2) - ((2*(a^3*f^3 - b^3*e^3 + 3*a*b^2*e^2*f - 3*a^2*b*e*f^2))/(7*(c*f - d*e)) - (2*(e + f*x)^3*(a^3
*d^3*f^3 - b^3*c^3*f^3 + 3*a*b^2*c^2*d*f^3 - 3*a^2*b*c*d^2*f^3))/(c*f - d*e)^4 + (2*(e + f*x)^2*(a^3*d^2*f^3 -
b^3*d^2*e^3 + 3*a*b^2*c^2*f^3 - 3*b^3*c^2*e*f^2 - 3*a^2*b*c*d*f^3 + 3*b^3*c*d*e^2*f))/(3*(c*f - d*e)^3) - (2*
(e + f*x)*(a^3*d*f^3 + 2*b^3*d*e^3 - 3*a^2*b*c*f^3 - 3*b^3*c*e^2*f + 6*a*b^2*c*e*f^2 - 3*a*b^2*d*e^2*f))/(5*(c
*f - d*e)^2))/(f^3*(e + f*x)^(7/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(9/2),x)

[Out]

Timed out

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