3.18.81 \(\int \frac {(a+b x)^2}{(c+d x) (e+f x)^{9/2}} \, dx\) [1781]

Optimal. Leaf size=207 \[ \frac {2 (b e-a f)^2}{7 f^2 (d e-c f) (e+f x)^{7/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{5 f^2 (d e-c f)^2 (e+f x)^{5/2}}+\frac {2 (b c-a d)^2}{3 (d e-c f)^3 (e+f x)^{3/2}}+\frac {2 d (b c-a d)^2}{(d e-c f)^4 \sqrt {e+f x}}-\frac {2 d^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 207, 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 = {89, 65, 214} \begin {gather*} -\frac {2 d^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}}-\frac {2 (b e-a f) (a d f-2 b c f+b d e)}{5 f^2 (e+f x)^{5/2} (d e-c f)^2}+\frac {2 (b e-a f)^2}{7 f^2 (e+f x)^{7/2} (d e-c f)}+\frac {2 d (b c-a d)^2}{\sqrt {e+f x} (d e-c f)^4}+\frac {2 (b c-a d)^2}{3 (e+f x)^{3/2} (d e-c f)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 65

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 89

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{9/2}} \, dx &=\int \left (\frac {(-b e+a f)^2}{f (-d e+c f) (e+f x)^{9/2}}+\frac {(-b e+a f) (-b d e+2 b c f-a d f)}{f (-d e+c f)^2 (e+f x)^{7/2}}+\frac {(b c-a d)^2 f}{(-d e+c f)^3 (e+f x)^{5/2}}-\frac {d (-b c+a d)^2 f}{(-d e+c f)^4 (e+f x)^{3/2}}+\frac {d^2 (-b c+a d)^2}{(d e-c f)^4 (c+d x) \sqrt {e+f x}}\right ) \, dx\\ &=\frac {2 (b e-a f)^2}{7 f^2 (d e-c f) (e+f x)^{7/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{5 f^2 (d e-c f)^2 (e+f x)^{5/2}}+\frac {2 (b c-a d)^2}{3 (d e-c f)^3 (e+f x)^{3/2}}+\frac {2 d (b c-a d)^2}{(d e-c f)^4 \sqrt {e+f x}}+\frac {\left (d^2 (b c-a d)^2\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{(d e-c f)^4}\\ &=\frac {2 (b e-a f)^2}{7 f^2 (d e-c f) (e+f x)^{7/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{5 f^2 (d e-c f)^2 (e+f x)^{5/2}}+\frac {2 (b c-a d)^2}{3 (d e-c f)^3 (e+f x)^{3/2}}+\frac {2 d (b c-a d)^2}{(d e-c f)^4 \sqrt {e+f x}}+\frac {\left (2 d^2 (b c-a d)^2\right ) \text {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)^2}{7 f^2 (d e-c f) (e+f x)^{7/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{5 f^2 (d e-c f)^2 (e+f x)^{5/2}}+\frac {2 (b c-a d)^2}{3 (d e-c f)^3 (e+f x)^{3/2}}+\frac {2 d (b c-a d)^2}{(d e-c f)^4 \sqrt {e+f x}}-\frac {2 d^{3/2} (b c-a d)^2 \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 [A]
time = 0.65, size = 378, normalized size = 1.83 \begin {gather*} \frac {-2 b^2 \left (3 d^3 e^4 (2 e+7 f x)-3 c d^2 e^3 f (13 e+28 f x)+c^3 f^3 \left (8 e^2+28 e f x+35 f^2 x^2\right )-5 c^2 d f^2 \left (16 e^3+56 e^2 f x+70 e f^2 x^2+21 f^3 x^3\right )\right )-4 a b f \left (15 d^3 e^4+3 c^3 f^3 (2 e+7 f x)-c^2 d f^2 \left (32 e^2+112 e f x+35 f^2 x^2\right )+c d^2 f \left (116 e^3+406 e^2 f x+350 e f^2 x^2+105 f^3 x^3\right )\right )+2 a^2 f^2 \left (-15 c^3 f^3+3 c^2 d f^2 (22 e+7 f x)-c d^2 f \left (122 e^2+112 e f x+35 f^2 x^2\right )+d^3 \left (176 e^3+406 e^2 f x+350 e f^2 x^2+105 f^3 x^3\right )\right )}{105 f^2 (d e-c f)^4 (e+f x)^{7/2}}+\frac {2 d^{3/2} (b c-a d)^2 \tan ^{-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]

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

[Out]

(-2*b^2*(3*d^3*e^4*(2*e + 7*f*x) - 3*c*d^2*e^3*f*(13*e + 28*f*x) + c^3*f^3*(8*e^2 + 28*e*f*x + 35*f^2*x^2) - 5
*c^2*d*f^2*(16*e^3 + 56*e^2*f*x + 70*e*f^2*x^2 + 21*f^3*x^3)) - 4*a*b*f*(15*d^3*e^4 + 3*c^3*f^3*(2*e + 7*f*x)
- c^2*d*f^2*(32*e^2 + 112*e*f*x + 35*f^2*x^2) + c*d^2*f*(116*e^3 + 406*e^2*f*x + 350*e*f^2*x^2 + 105*f^3*x^3))
+ 2*a^2*f^2*(-15*c^3*f^3 + 3*c^2*d*f^2*(22*e + 7*f*x) - c*d^2*f*(122*e^2 + 112*e*f*x + 35*f^2*x^2) + d^3*(176
*e^3 + 406*e^2*f*x + 350*e*f^2*x^2 + 105*f^3*x^3)))/(105*f^2*(d*e - c*f)^4*(e + f*x)^(7/2)) + (2*d^(3/2)*(b*c
- a*d)^2*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(-(d*e) + c*f)^(9/2)

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Maple [A]
time = 0.11, size = 258, normalized size = 1.25

method	result	size
derivativedivides	\(\frac {\frac {2 d^{2} f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}}}-\frac {2 \left (-a^{2} d \,f^{2}+2 a b c \,f^{2}-2 b^{2} c e f +b^{2} d \,e^{2}\right )}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}-\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}{\left (c f -d e \right )^{4} \sqrt {f x +e}}}{f^{2}}\)	\(258\)
default	\(\frac {\frac {2 d^{2} f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}}}-\frac {2 \left (-a^{2} d \,f^{2}+2 a b c \,f^{2}-2 b^{2} c e f +b^{2} d \,e^{2}\right )}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}-\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}{\left (c f -d e \right )^{4} \sqrt {f x +e}}}{f^{2}}\)	\(258\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^2/(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-%e*d>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (199) = 398\).
time = 1.40, size = 1804, normalized size = 8.71 \begin {gather*} \left [\frac {105 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{6} x^{4} + 4 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{5} x^{3} e + 6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{4} x^{2} e^{2} + 4 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3} x e^{3} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{2} e^{4}\right )} \sqrt {-\frac {d}{c f - d e}} \log \left (\frac {d f x - c f + 2 \, {\left (c f - d e\right )} \sqrt {f x + e} \sqrt {-\frac {d}{c f - d e}} + 2 \, d e}{d x + c}\right ) - 2 \, {\left (15 \, a^{2} c^{3} f^{5} - 105 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{5} x^{3} + 35 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{5} x^{2} + 21 \, {\left (2 \, a b c^{3} - a^{2} c^{2} d\right )} f^{5} x + 6 \, b^{2} d^{3} e^{5} + 3 \, {\left (7 \, b^{2} d^{3} f x - {\left (13 \, b^{2} c d^{2} - 10 \, a b d^{3}\right )} f\right )} e^{4} - 4 \, {\left (21 \, b^{2} c d^{2} f^{2} x + 2 \, {\left (10 \, b^{2} c^{2} d - 29 \, a b c d^{2} + 22 \, a^{2} d^{3}\right )} f^{2}\right )} e^{3} - 2 \, {\left (7 \, {\left (20 \, b^{2} c^{2} d - 58 \, a b c d^{2} + 29 \, a^{2} d^{3}\right )} f^{3} x - {\left (4 \, b^{2} c^{3} - 32 \, a b c^{2} d + 61 \, a^{2} c d^{2}\right )} f^{3}\right )} e^{2} - 2 \, {\left (175 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{4} x^{2} - 14 \, {\left (b^{2} c^{3} - 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} f^{4} x - 3 \, {\left (2 \, a b c^{3} - 11 \, a^{2} c^{2} d\right )} f^{4}\right )} e\right )} \sqrt {f x + e}}{105 \, {\left (c^{4} f^{10} x^{4} + d^{4} f^{2} e^{8} + 4 \, {\left (d^{4} f^{3} x - c d^{3} f^{3}\right )} e^{7} + 2 \, {\left (3 \, d^{4} f^{4} x^{2} - 8 \, c d^{3} f^{4} x + 3 \, c^{2} d^{2} f^{4}\right )} e^{6} + 4 \, {\left (d^{4} f^{5} x^{3} - 6 \, c d^{3} f^{5} x^{2} + 6 \, c^{2} d^{2} f^{5} x - c^{3} d f^{5}\right )} e^{5} + {\left (d^{4} f^{6} x^{4} - 16 \, c d^{3} f^{6} x^{3} + 36 \, c^{2} d^{2} f^{6} x^{2} - 16 \, c^{3} d f^{6} x + c^{4} f^{6}\right )} e^{4} - 4 \, {\left (c d^{3} f^{7} x^{4} - 6 \, c^{2} d^{2} f^{7} x^{3} + 6 \, c^{3} d f^{7} x^{2} - c^{4} f^{7} x\right )} e^{3} + 2 \, {\left (3 \, c^{2} d^{2} f^{8} x^{4} - 8 \, c^{3} d f^{8} x^{3} + 3 \, c^{4} f^{8} x^{2}\right )} e^{2} - 4 \, {\left (c^{3} d f^{9} x^{4} - c^{4} f^{9} x^{3}\right )} e\right )}}, \frac {2 \, {\left (105 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{6} x^{4} + 4 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{5} x^{3} e + 6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{4} x^{2} e^{2} + 4 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3} x e^{3} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{2} e^{4}\right )} \sqrt {\frac {d}{c f - d e}} \arctan \left (-\frac {{\left (c f - d e\right )} \sqrt {f x + e} \sqrt {\frac {d}{c f - d e}}}{d f x + d e}\right ) - {\left (15 \, a^{2} c^{3} f^{5} - 105 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{5} x^{3} + 35 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{5} x^{2} + 21 \, {\left (2 \, a b c^{3} - a^{2} c^{2} d\right )} f^{5} x + 6 \, b^{2} d^{3} e^{5} + 3 \, {\left (7 \, b^{2} d^{3} f x - {\left (13 \, b^{2} c d^{2} - 10 \, a b d^{3}\right )} f\right )} e^{4} - 4 \, {\left (21 \, b^{2} c d^{2} f^{2} x + 2 \, {\left (10 \, b^{2} c^{2} d - 29 \, a b c d^{2} + 22 \, a^{2} d^{3}\right )} f^{2}\right )} e^{3} - 2 \, {\left (7 \, {\left (20 \, b^{2} c^{2} d - 58 \, a b c d^{2} + 29 \, a^{2} d^{3}\right )} f^{3} x - {\left (4 \, b^{2} c^{3} - 32 \, a b c^{2} d + 61 \, a^{2} c d^{2}\right )} f^{3}\right )} e^{2} - 2 \, {\left (175 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{4} x^{2} - 14 \, {\left (b^{2} c^{3} - 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} f^{4} x - 3 \, {\left (2 \, a b c^{3} - 11 \, a^{2} c^{2} d\right )} f^{4}\right )} e\right )} \sqrt {f x + e}\right )}}{105 \, {\left (c^{4} f^{10} x^{4} + d^{4} f^{2} e^{8} + 4 \, {\left (d^{4} f^{3} x - c d^{3} f^{3}\right )} e^{7} + 2 \, {\left (3 \, d^{4} f^{4} x^{2} - 8 \, c d^{3} f^{4} x + 3 \, c^{2} d^{2} f^{4}\right )} e^{6} + 4 \, {\left (d^{4} f^{5} x^{3} - 6 \, c d^{3} f^{5} x^{2} + 6 \, c^{2} d^{2} f^{5} x - c^{3} d f^{5}\right )} e^{5} + {\left (d^{4} f^{6} x^{4} - 16 \, c d^{3} f^{6} x^{3} + 36 \, c^{2} d^{2} f^{6} x^{2} - 16 \, c^{3} d f^{6} x + c^{4} f^{6}\right )} e^{4} - 4 \, {\left (c d^{3} f^{7} x^{4} - 6 \, c^{2} d^{2} f^{7} x^{3} + 6 \, c^{3} d f^{7} x^{2} - c^{4} f^{7} x\right )} e^{3} + 2 \, {\left (3 \, c^{2} d^{2} f^{8} x^{4} - 8 \, c^{3} d f^{8} x^{3} + 3 \, c^{4} f^{8} x^{2}\right )} e^{2} - 4 \, {\left (c^{3} d f^{9} x^{4} - c^{4} f^{9} x^{3}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/105*(105*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^6*x^4 + 4*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^5*x^3*e + 6
*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^4*x^2*e^2 + 4*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^3*x*e^3 + (b^2*c^2*
d - 2*a*b*c*d^2 + a^2*d^3)*f^2*e^4)*sqrt(-d/(c*f - d*e))*log((d*f*x - c*f + 2*(c*f - d*e)*sqrt(f*x + e)*sqrt(-
d/(c*f - d*e)) + 2*d*e)/(d*x + c)) - 2*(15*a^2*c^3*f^5 - 105*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^5*x^3 + 35*
(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^5*x^2 + 21*(2*a*b*c^3 - a^2*c^2*d)*f^5*x + 6*b^2*d^3*e^5 + 3*(7*b^2*d^3*
f*x - (13*b^2*c*d^2 - 10*a*b*d^3)*f)*e^4 - 4*(21*b^2*c*d^2*f^2*x + 2*(10*b^2*c^2*d - 29*a*b*c*d^2 + 22*a^2*d^3
)*f^2)*e^3 - 2*(7*(20*b^2*c^2*d - 58*a*b*c*d^2 + 29*a^2*d^3)*f^3*x - (4*b^2*c^3 - 32*a*b*c^2*d + 61*a^2*c*d^2)
*f^3)*e^2 - 2*(175*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^4*x^2 - 14*(b^2*c^3 - 8*a*b*c^2*d + 4*a^2*c*d^2)*f^4*
x - 3*(2*a*b*c^3 - 11*a^2*c^2*d)*f^4)*e)*sqrt(f*x + e))/(c^4*f^10*x^4 + d^4*f^2*e^8 + 4*(d^4*f^3*x - c*d^3*f^3
)*e^7 + 2*(3*d^4*f^4*x^2 - 8*c*d^3*f^4*x + 3*c^2*d^2*f^4)*e^6 + 4*(d^4*f^5*x^3 - 6*c*d^3*f^5*x^2 + 6*c^2*d^2*f
^5*x - c^3*d*f^5)*e^5 + (d^4*f^6*x^4 - 16*c*d^3*f^6*x^3 + 36*c^2*d^2*f^6*x^2 - 16*c^3*d*f^6*x + c^4*f^6)*e^4 -
4*(c*d^3*f^7*x^4 - 6*c^2*d^2*f^7*x^3 + 6*c^3*d*f^7*x^2 - c^4*f^7*x)*e^3 + 2*(3*c^2*d^2*f^8*x^4 - 8*c^3*d*f^8*
x^3 + 3*c^4*f^8*x^2)*e^2 - 4*(c^3*d*f^9*x^4 - c^4*f^9*x^3)*e), 2/105*(105*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)
*f^6*x^4 + 4*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^5*x^3*e + 6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^4*x^2*e^2
+ 4*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^3*x*e^3 + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^2*e^4)*sqrt(d/(c*f
- d*e))*arctan(-(c*f - d*e)*sqrt(f*x + e)*sqrt(d/(c*f - d*e))/(d*f*x + d*e)) - (15*a^2*c^3*f^5 - 105*(b^2*c^2*
d - 2*a*b*c*d^2 + a^2*d^3)*f^5*x^3 + 35*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^5*x^2 + 21*(2*a*b*c^3 - a^2*c^2*
d)*f^5*x + 6*b^2*d^3*e^5 + 3*(7*b^2*d^3*f*x - (13*b^2*c*d^2 - 10*a*b*d^3)*f)*e^4 - 4*(21*b^2*c*d^2*f^2*x + 2*(
10*b^2*c^2*d - 29*a*b*c*d^2 + 22*a^2*d^3)*f^2)*e^3 - 2*(7*(20*b^2*c^2*d - 58*a*b*c*d^2 + 29*a^2*d^3)*f^3*x - (
4*b^2*c^3 - 32*a*b*c^2*d + 61*a^2*c*d^2)*f^3)*e^2 - 2*(175*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^4*x^2 - 14*(b
^2*c^3 - 8*a*b*c^2*d + 4*a^2*c*d^2)*f^4*x - 3*(2*a*b*c^3 - 11*a^2*c^2*d)*f^4)*e)*sqrt(f*x + e))/(c^4*f^10*x^4
+ d^4*f^2*e^8 + 4*(d^4*f^3*x - c*d^3*f^3)*e^7 + 2*(3*d^4*f^4*x^2 - 8*c*d^3*f^4*x + 3*c^2*d^2*f^4)*e^6 + 4*(d^4
*f^5*x^3 - 6*c*d^3*f^5*x^2 + 6*c^2*d^2*f^5*x - c^3*d*f^5)*e^5 + (d^4*f^6*x^4 - 16*c*d^3*f^6*x^3 + 36*c^2*d^2*f
^6*x^2 - 16*c^3*d*f^6*x + c^4*f^6)*e^4 - 4*(c*d^3*f^7*x^4 - 6*c^2*d^2*f^7*x^3 + 6*c^3*d*f^7*x^2 - c^4*f^7*x)*e
^3 + 2*(3*c^2*d^2*f^8*x^4 - 8*c^3*d*f^8*x^3 + 3*c^4*f^8*x^2)*e^2 - 4*(c^3*d*f^9*x^4 - c^4*f^9*x^3)*e)]

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Sympy [A]
time = 136.61, size = 189, normalized size = 0.91 \begin {gather*} \frac {2 d \left (a d - b c\right )^{2}}{\sqrt {e + f x} \left (c f - d e\right )^{4}} + \frac {2 d \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{4}} - \frac {2 \left (a d - b c\right )^{2}}{3 \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )^{3}} + \frac {2 \left (a f - b e\right ) \left (a d f - 2 b c f + b d e\right )}{5 f^{2} \left (e + f x\right )^{\frac {5}{2}} \left (c f - d e\right )^{2}} - \frac {2 \left (a f - b e\right )^{2}}{7 f^{2} \left (e + f x\right )^{\frac {7}{2}} \left (c f - d e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d*(a*d - b*c)**2/(sqrt(e + f*x)*(c*f - d*e)**4) + 2*d*(a*d - b*c)**2*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))
/(sqrt((c*f - d*e)/d)*(c*f - d*e)**4) - 2*(a*d - b*c)**2/(3*(e + f*x)**(3/2)*(c*f - d*e)**3) + 2*(a*f - b*e)*(
a*d*f - 2*b*c*f + b*d*e)/(5*f**2*(e + f*x)**(5/2)*(c*f - d*e)**2) - 2*(a*f - b*e)**2/(7*f**2*(e + f*x)**(7/2)*
(c*f - d*e))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (199) = 398\).
time = 0.61, size = 695, normalized size = 3.36 \begin {gather*} \frac {2 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} 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^{2} c^{2} d f^{2} - 210 \, {\left (f x + e\right )}^{3} a b c d^{2} f^{2} + 105 \, {\left (f x + e\right )}^{3} a^{2} d^{3} f^{2} - 35 \, {\left (f x + e\right )}^{2} b^{2} c^{3} f^{3} + 70 \, {\left (f x + e\right )}^{2} a b c^{2} d f^{3} - 35 \, {\left (f x + e\right )}^{2} a^{2} c d^{2} f^{3} - 42 \, {\left (f x + e\right )} a b c^{3} f^{4} + 21 \, {\left (f x + e\right )} a^{2} c^{2} d f^{4} - 15 \, a^{2} c^{3} f^{5} + 35 \, {\left (f x + e\right )}^{2} b^{2} c^{2} d f^{2} e - 70 \, {\left (f x + e\right )}^{2} a b c d^{2} f^{2} e + 35 \, {\left (f x + e\right )}^{2} a^{2} d^{3} f^{2} e + 42 \, {\left (f x + e\right )} b^{2} c^{3} f^{3} e + 84 \, {\left (f x + e\right )} a b c^{2} d f^{3} e - 42 \, {\left (f x + e\right )} a^{2} c d^{2} f^{3} e + 30 \, a b c^{3} f^{4} e + 45 \, a^{2} c^{2} d f^{4} e - 105 \, {\left (f x + e\right )} b^{2} c^{2} d f^{2} e^{2} - 42 \, {\left (f x + e\right )} a b c d^{2} f^{2} e^{2} + 21 \, {\left (f x + e\right )} a^{2} d^{3} f^{2} e^{2} - 15 \, b^{2} c^{3} f^{3} e^{2} - 90 \, a b c^{2} d f^{3} e^{2} - 45 \, a^{2} c d^{2} f^{3} e^{2} + 84 \, {\left (f x + e\right )} b^{2} c d^{2} f e^{3} + 45 \, b^{2} c^{2} d f^{2} e^{3} + 90 \, a b c d^{2} f^{2} e^{3} + 15 \, a^{2} d^{3} f^{2} e^{3} - 21 \, {\left (f x + e\right )} b^{2} d^{3} e^{4} - 45 \, b^{2} c d^{2} f e^{4} - 30 \, a b d^{3} f e^{4} + 15 \, b^{2} d^{3} e^{5}\right )}}{105 \, {\left (c^{4} f^{6} - 4 \, c^{3} d f^{5} e + 6 \, c^{2} d^{2} f^{4} e^{2} - 4 \, c d^{3} f^{3} e^{3} + d^{4} f^{2} 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)^2/(d*x+c)/(f*x+e)^(9/2),x, algorithm="giac")

[Out]

2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*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^2*c^2*d*f^2 - 2
10*(f*x + e)^3*a*b*c*d^2*f^2 + 105*(f*x + e)^3*a^2*d^3*f^2 - 35*(f*x + e)^2*b^2*c^3*f^3 + 70*(f*x + e)^2*a*b*c
^2*d*f^3 - 35*(f*x + e)^2*a^2*c*d^2*f^3 - 42*(f*x + e)*a*b*c^3*f^4 + 21*(f*x + e)*a^2*c^2*d*f^4 - 15*a^2*c^3*f
^5 + 35*(f*x + e)^2*b^2*c^2*d*f^2*e - 70*(f*x + e)^2*a*b*c*d^2*f^2*e + 35*(f*x + e)^2*a^2*d^3*f^2*e + 42*(f*x
+ e)*b^2*c^3*f^3*e + 84*(f*x + e)*a*b*c^2*d*f^3*e - 42*(f*x + e)*a^2*c*d^2*f^3*e + 30*a*b*c^3*f^4*e + 45*a^2*c
^2*d*f^4*e - 105*(f*x + e)*b^2*c^2*d*f^2*e^2 - 42*(f*x + e)*a*b*c*d^2*f^2*e^2 + 21*(f*x + e)*a^2*d^3*f^2*e^2 -
15*b^2*c^3*f^3*e^2 - 90*a*b*c^2*d*f^3*e^2 - 45*a^2*c*d^2*f^3*e^2 + 84*(f*x + e)*b^2*c*d^2*f*e^3 + 45*b^2*c^2*
d*f^2*e^3 + 90*a*b*c*d^2*f^2*e^3 + 15*a^2*d^3*f^2*e^3 - 21*(f*x + e)*b^2*d^3*e^4 - 45*b^2*c*d^2*f*e^4 - 30*a*b
*d^3*f*e^4 + 15*b^2*d^3*e^5)/((c^4*f^6 - 4*c^3*d*f^5*e + 6*c^2*d^2*f^4*e^2 - 4*c*d^3*f^3*e^3 + d^4*f^2*e^4)*(f
*x + e)^(7/2))

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Mupad [B]
time = 1.47, size = 327, normalized size = 1.58 \begin {gather*} \frac {2\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,\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^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (c\,f-d\,e\right )}^{9/2}}-\frac {\frac {2\,\left (a^2\,f^2-2\,a\,b\,e\,f+b^2\,e^2\right )}{7\,\left (c\,f-d\,e\right )}+\frac {2\,{\left (e+f\,x\right )}^2\,\left (a^2\,d^2\,f^2-2\,a\,b\,c\,d\,f^2+b^2\,c^2\,f^2\right )}{3\,{\left (c\,f-d\,e\right )}^3}-\frac {2\,\left (e+f\,x\right )\,\left (d\,a^2\,f^2-2\,c\,a\,b\,f^2-d\,b^2\,e^2+2\,c\,b^2\,e\,f\right )}{5\,{\left (c\,f-d\,e\right )}^2}-\frac {2\,d\,{\left (e+f\,x\right )}^3\,\left (a^2\,d^2\,f^2-2\,a\,b\,c\,d\,f^2+b^2\,c^2\,f^2\right )}{{\left (c\,f-d\,e\right )}^4}}{f^2\,{\left (e+f\,x\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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