3.18.87 \(\int \frac {(a+b x)^3}{(c+d x) (e+f x)^{5/2}} \, dx\) [1787]

Optimal. Leaf size=163 \[ -\frac {2 (b e-a f)^3}{3 f^3 (d e-c f) (e+f x)^{3/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{f^3 (d e-c f)^2 \sqrt {e+f x}}+\frac {2 b^3 \sqrt {e+f x}}{d f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 163, 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 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}}+\frac {2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{f^3 \sqrt {e+f x} (d e-c f)^2}-\frac {2 (b e-a f)^3}{3 f^3 (e+f x)^{3/2} (d e-c f)}+\frac {2 b^3 \sqrt {e+f x}}{d f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/((c + d*x)*(e + f*x)^(5/2)),x]

[Out]

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

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)^3}{(c+d x) (e+f x)^{5/2}} \, dx &=\int \left (\frac {(-b e+a f)^3}{f^2 (-d e+c f) (e+f x)^{5/2}}+\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)^{3/2}}+\frac {b^3}{d f^2 \sqrt {e+f x}}+\frac {(-b c+a d)^3}{d (d e-c f)^2 (c+d x) \sqrt {e+f x}}\right ) \, dx\\ &=-\frac {2 (b e-a f)^3}{3 f^3 (d e-c f) (e+f x)^{3/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{f^3 (d e-c f)^2 \sqrt {e+f x}}+\frac {2 b^3 \sqrt {e+f x}}{d f^3}-\frac {(b c-a d)^3 \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d (d e-c f)^2}\\ &=-\frac {2 (b e-a f)^3}{3 f^3 (d e-c f) (e+f x)^{3/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{f^3 (d e-c f)^2 \sqrt {e+f x}}+\frac {2 b^3 \sqrt {e+f x}}{d f^3}-\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d f (d e-c f)^2}\\ &=-\frac {2 (b e-a f)^3}{3 f^3 (d e-c f) (e+f x)^{3/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{f^3 (d e-c f)^2 \sqrt {e+f x}}+\frac {2 b^3 \sqrt {e+f x}}{d f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 241, normalized size = 1.48 \begin {gather*} \frac {2 \left (a^3 d f^3 (4 d e-c f+3 d f x)-3 a^2 b d f^2 \left (d e^2+c f (2 e+3 f x)\right )-3 a b^2 d e f (d e (2 e+3 f x)-c f (5 e+6 f x))+b^3 \left (3 c^2 f^2 (e+f x)^2+d^2 e^2 \left (8 e^2+12 e f x+3 f^2 x^2\right )-c d e f \left (14 e^2+21 e f x+6 f^2 x^2\right )\right )\right )}{3 d f^3 (d e-c f)^2 (e+f x)^{3/2}}+\frac {2 (-b c+a d)^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2} (-d e+c f)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/((c + d*x)*(e + f*x)^(5/2)),x]

[Out]

(2*(a^3*d*f^3*(4*d*e - c*f + 3*d*f*x) - 3*a^2*b*d*f^2*(d*e^2 + c*f*(2*e + 3*f*x)) - 3*a*b^2*d*e*f*(d*e*(2*e +
3*f*x) - c*f*(5*e + 6*f*x)) + b^3*(3*c^2*f^2*(e + f*x)^2 + d^2*e^2*(8*e^2 + 12*e*f*x + 3*f^2*x^2) - c*d*e*f*(1
4*e^2 + 21*e*f*x + 6*f^2*x^2))))/(3*d*f^3*(d*e - c*f)^2*(e + f*x)^(3/2)) + (2*(-(b*c) + a*d)^3*ArcTan[(Sqrt[d]
*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(3/2)*(-(d*e) + c*f)^(5/2))

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Maple [A]
time = 0.14, size = 243, normalized size = 1.49

method	result	size
derivativedivides	\(\frac {\frac {2 b^{3} \sqrt {f x +e}}{d}+\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d \left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}-\frac {2 \left (-a^{3} d \,f^{3}+3 a^{2} b c \,f^{3}-6 a \,b^{2} c e \,f^{2}+3 a \,b^{2} d \,e^{2} f +3 b^{3} c \,e^{2} f -2 b^{3} d \,e^{3}\right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f^{3}}\)	\(243\)
default	\(\frac {\frac {2 b^{3} \sqrt {f x +e}}{d}+\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d \left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}-\frac {2 \left (-a^{3} d \,f^{3}+3 a^{2} b c \,f^{3}-6 a \,b^{2} c e \,f^{2}+3 a \,b^{2} d \,e^{2} f +3 b^{3} c \,e^{2} f -2 b^{3} d \,e^{3}\right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f^{3}}\)	\(243\)
risch	\(\frac {2 b^{3} \sqrt {f x +e}}{d \,f^{3}}-\frac {2 a^{3}}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 a^{2} b e}{f \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}-\frac {2 a \,b^{2} e^{2}}{f^{2} \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 b^{3} e^{3}}{3 f^{3} \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 d \,a^{3}}{\left (c f -d e \right )^{2} \sqrt {f x +e}}-\frac {6 a^{2} b c}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {12 a \,b^{2} c e}{f \left (c f -d e \right )^{2} \sqrt {f x +e}}-\frac {6 d a \,b^{2} e^{2}}{f^{2} \left (c f -d e \right )^{2} \sqrt {f x +e}}-\frac {6 b^{3} c \,e^{2}}{f^{2} \left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {4 d \,b^{3} e^{3}}{f^{3} \left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {2 d^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{3}}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {6 d \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} b c}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}+\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a \,b^{2} c^{2}}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{3} c^{3}}{d \left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}\)	\(501\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)/(f*x+e)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/f^3*(b^3/d*(f*x+e)^(1/2)+f^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d/(c*f-d*e)^2/((c*f-d*e)*d)^(1/2)
*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))-1/3*(a^3*f^3-3*a^2*b*e*f^2+3*a*b^2*e^2*f-b^3*e^3)/(c*f-d*e)/(f*x+
e)^(3/2)-1/(c*f-d*e)^2*(-a^3*d*f^3+3*a^2*b*c*f^3-6*a*b^2*c*e*f^2+3*a*b^2*d*e^2*f+3*b^3*c*e^2*f-2*b^3*d*e^3)/(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)^3/(d*x+c)/(f*x+e)^(5/2),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. 714 vs. \(2 (157) = 314\).
time = 1.22, size = 1442, normalized size = 8.85 \begin {gather*} \left [\frac {3 \, {\left ({\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{5} x^{2} + 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{4} x e + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} e^{2}\right )} \sqrt {-c d f + d^{2} e} \log \left (\frac {d f x - c f + 2 \, d e - 2 \, \sqrt {-c d f + d^{2} e} \sqrt {f x + e}}{d x + c}\right ) + 2 \, {\left (3 \, b^{3} c^{3} d f^{5} x^{2} - a^{3} c^{2} d^{2} f^{5} - 8 \, b^{3} d^{4} e^{5} - 3 \, {\left (3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} f^{5} x - 2 \, {\left (6 \, b^{3} d^{4} f x - {\left (11 \, b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} f\right )} e^{4} - {\left (3 \, b^{3} d^{4} f^{2} x^{2} - 3 \, {\left (11 \, b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} f^{2} x + {\left (17 \, b^{3} c^{2} d^{2} + 21 \, a b^{2} c d^{3} - 3 \, a^{2} b d^{4}\right )} f^{2}\right )} e^{3} + {\left (9 \, b^{3} c d^{3} f^{3} x^{2} - 27 \, {\left (b^{3} c^{2} d^{2} + a b^{2} c d^{3}\right )} f^{3} x + {\left (3 \, b^{3} c^{3} d + 15 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 4 \, a^{3} d^{4}\right )} f^{3}\right )} e^{2} - {\left (9 \, b^{3} c^{2} d^{2} f^{4} x^{2} - 3 \, {\left (2 \, b^{3} c^{3} d + 6 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{4} x + {\left (6 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} f^{4}\right )} e\right )} \sqrt {f x + e}}{3 \, {\left (c^{3} d^{2} f^{8} x^{2} - d^{5} f^{3} e^{5} - {\left (2 \, d^{5} f^{4} x - 3 \, c d^{4} f^{4}\right )} e^{4} - {\left (d^{5} f^{5} x^{2} - 6 \, c d^{4} f^{5} x + 3 \, c^{2} d^{3} f^{5}\right )} e^{3} + {\left (3 \, c d^{4} f^{6} x^{2} - 6 \, c^{2} d^{3} f^{6} x + c^{3} d^{2} f^{6}\right )} e^{2} - {\left (3 \, c^{2} d^{3} f^{7} x^{2} - 2 \, c^{3} d^{2} f^{7} x\right )} e\right )}}, \frac {2 \, {\left (3 \, {\left ({\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{5} x^{2} + 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{4} x e + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} e^{2}\right )} \sqrt {c d f - d^{2} e} \arctan \left (\frac {\sqrt {c d f - d^{2} e} \sqrt {f x + e}}{d f x + d e}\right ) + {\left (3 \, b^{3} c^{3} d f^{5} x^{2} - a^{3} c^{2} d^{2} f^{5} - 8 \, b^{3} d^{4} e^{5} - 3 \, {\left (3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} f^{5} x - 2 \, {\left (6 \, b^{3} d^{4} f x - {\left (11 \, b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} f\right )} e^{4} - {\left (3 \, b^{3} d^{4} f^{2} x^{2} - 3 \, {\left (11 \, b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} f^{2} x + {\left (17 \, b^{3} c^{2} d^{2} + 21 \, a b^{2} c d^{3} - 3 \, a^{2} b d^{4}\right )} f^{2}\right )} e^{3} + {\left (9 \, b^{3} c d^{3} f^{3} x^{2} - 27 \, {\left (b^{3} c^{2} d^{2} + a b^{2} c d^{3}\right )} f^{3} x + {\left (3 \, b^{3} c^{3} d + 15 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 4 \, a^{3} d^{4}\right )} f^{3}\right )} e^{2} - {\left (9 \, b^{3} c^{2} d^{2} f^{4} x^{2} - 3 \, {\left (2 \, b^{3} c^{3} d + 6 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{4} x + {\left (6 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} f^{4}\right )} e\right )} \sqrt {f x + e}\right )}}{3 \, {\left (c^{3} d^{2} f^{8} x^{2} - d^{5} f^{3} e^{5} - {\left (2 \, d^{5} f^{4} x - 3 \, c d^{4} f^{4}\right )} e^{4} - {\left (d^{5} f^{5} x^{2} - 6 \, c d^{4} f^{5} x + 3 \, c^{2} d^{3} f^{5}\right )} e^{3} + {\left (3 \, c d^{4} f^{6} x^{2} - 6 \, c^{2} d^{3} f^{6} x + c^{3} d^{2} f^{6}\right )} e^{2} - {\left (3 \, c^{2} d^{3} f^{7} x^{2} - 2 \, c^{3} d^{2} f^{7} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)/(f*x+e)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^5*x^2 + 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*
d^2 - a^3*d^3)*f^4*x*e + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*e^2)*sqrt(-c*d*f + d^2*e)*log
((d*f*x - c*f + 2*d*e - 2*sqrt(-c*d*f + d^2*e)*sqrt(f*x + e))/(d*x + c)) + 2*(3*b^3*c^3*d*f^5*x^2 - a^3*c^2*d^
2*f^5 - 8*b^3*d^4*e^5 - 3*(3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^5*x - 2*(6*b^3*d^4*f*x - (11*b^3*c*d^3 + 3*a*b^2*d^4
)*f)*e^4 - (3*b^3*d^4*f^2*x^2 - 3*(11*b^3*c*d^3 + 3*a*b^2*d^4)*f^2*x + (17*b^3*c^2*d^2 + 21*a*b^2*c*d^3 - 3*a^
2*b*d^4)*f^2)*e^3 + (9*b^3*c*d^3*f^3*x^2 - 27*(b^3*c^2*d^2 + a*b^2*c*d^3)*f^3*x + (3*b^3*c^3*d + 15*a*b^2*c^2*
d^2 + 3*a^2*b*c*d^3 - 4*a^3*d^4)*f^3)*e^2 - (9*b^3*c^2*d^2*f^4*x^2 - 3*(2*b^3*c^3*d + 6*a*b^2*c^2*d^2 + 3*a^2*
b*c*d^3 - a^3*d^4)*f^4*x + (6*a^2*b*c^2*d^2 - 5*a^3*c*d^3)*f^4)*e)*sqrt(f*x + e))/(c^3*d^2*f^8*x^2 - d^5*f^3*e
^5 - (2*d^5*f^4*x - 3*c*d^4*f^4)*e^4 - (d^5*f^5*x^2 - 6*c*d^4*f^5*x + 3*c^2*d^3*f^5)*e^3 + (3*c*d^4*f^6*x^2 -
6*c^2*d^3*f^6*x + c^3*d^2*f^6)*e^2 - (3*c^2*d^3*f^7*x^2 - 2*c^3*d^2*f^7*x)*e), 2/3*(3*((b^3*c^3 - 3*a*b^2*c^2*
d + 3*a^2*b*c*d^2 - a^3*d^3)*f^5*x^2 + 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^4*x*e + (b^3*c^
3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*e^2)*sqrt(c*d*f - d^2*e)*arctan(sqrt(c*d*f - d^2*e)*sqrt(f*x
+ e)/(d*f*x + d*e)) + (3*b^3*c^3*d*f^5*x^2 - a^3*c^2*d^2*f^5 - 8*b^3*d^4*e^5 - 3*(3*a^2*b*c^2*d^2 - a^3*c*d^3)
*f^5*x - 2*(6*b^3*d^4*f*x - (11*b^3*c*d^3 + 3*a*b^2*d^4)*f)*e^4 - (3*b^3*d^4*f^2*x^2 - 3*(11*b^3*c*d^3 + 3*a*b
^2*d^4)*f^2*x + (17*b^3*c^2*d^2 + 21*a*b^2*c*d^3 - 3*a^2*b*d^4)*f^2)*e^3 + (9*b^3*c*d^3*f^3*x^2 - 27*(b^3*c^2*
d^2 + a*b^2*c*d^3)*f^3*x + (3*b^3*c^3*d + 15*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - 4*a^3*d^4)*f^3)*e^2 - (9*b^3*c^2*
d^2*f^4*x^2 - 3*(2*b^3*c^3*d + 6*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4*x + (6*a^2*b*c^2*d^2 - 5*a^3*c*d
^3)*f^4)*e)*sqrt(f*x + e))/(c^3*d^2*f^8*x^2 - d^5*f^3*e^5 - (2*d^5*f^4*x - 3*c*d^4*f^4)*e^4 - (d^5*f^5*x^2 - 6
*c*d^4*f^5*x + 3*c^2*d^3*f^5)*e^3 + (3*c*d^4*f^6*x^2 - 6*c^2*d^3*f^6*x + c^3*d^2*f^6)*e^2 - (3*c^2*d^3*f^7*x^2
- 2*c^3*d^2*f^7*x)*e)]

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Sympy [A]
time = 69.63, size = 153, normalized size = 0.94 \begin {gather*} \frac {2 b^{3} \sqrt {e + f x}}{d f^{3}} + \frac {2 \left (a f - b e\right )^{2} \left (a d f - 3 b c f + 2 b d e\right )}{f^{3} \sqrt {e + f x} \left (c f - d e\right )^{2}} - \frac {2 \left (a f - b e\right )^{3}}{3 f^{3} \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )} + \frac {2 \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{2} \sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{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)**(5/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (157) = 314\).
time = 0.68, size = 337, normalized size = 2.07 \begin {gather*} -\frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{{\left (c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2}\right )} \sqrt {c d f - d^{2} e}} + \frac {2 \, \sqrt {f x + e} b^{3}}{d f^{3}} - \frac {2 \, {\left (9 \, {\left (f x + e\right )} a^{2} b c f^{3} - 3 \, {\left (f x + e\right )} a^{3} d f^{3} + a^{3} c f^{4} - 18 \, {\left (f x + e\right )} a b^{2} c f^{2} e - 3 \, a^{2} b c f^{3} e - a^{3} d f^{3} e + 9 \, {\left (f x + e\right )} b^{3} c f e^{2} + 9 \, {\left (f x + e\right )} a b^{2} d f e^{2} + 3 \, a b^{2} c f^{2} e^{2} + 3 \, a^{2} b d f^{2} e^{2} - 6 \, {\left (f x + e\right )} b^{3} d e^{3} - b^{3} c f e^{3} - 3 \, a b^{2} d f e^{3} + b^{3} d e^{4}\right )}}{3 \, {\left (c^{2} f^{5} - 2 \, c d f^{4} e + d^{2} f^{3} e^{2}\right )} {\left (f x + e\right )}^{\frac {3}{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)^(5/2),x, algorithm="giac")

[Out]

-2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/((c^2*d*f^2
- 2*c*d^2*f*e + d^3*e^2)*sqrt(c*d*f - d^2*e)) + 2*sqrt(f*x + e)*b^3/(d*f^3) - 2/3*(9*(f*x + e)*a^2*b*c*f^3 -
3*(f*x + e)*a^3*d*f^3 + a^3*c*f^4 - 18*(f*x + e)*a*b^2*c*f^2*e - 3*a^2*b*c*f^3*e - a^3*d*f^3*e + 9*(f*x + e)*b
^3*c*f*e^2 + 9*(f*x + e)*a*b^2*d*f*e^2 + 3*a*b^2*c*f^2*e^2 + 3*a^2*b*d*f^2*e^2 - 6*(f*x + e)*b^3*d*e^3 - b^3*c
*f*e^3 - 3*a*b^2*d*f*e^3 + b^3*d*e^4)/((c^2*f^5 - 2*c*d*f^4*e + d^2*f^3*e^2)*(f*x + e)^(3/2))

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Mupad [B]
time = 0.21, size = 295, normalized size = 1.81 \begin {gather*} \frac {2\,b^3\,\sqrt {e+f\,x}}{d\,f^3}-\frac {\frac {2\,\left (d\,a^3\,f^3-3\,d\,a^2\,b\,e\,f^2+3\,d\,a\,b^2\,e^2\,f-d\,b^3\,e^3\right )}{3\,\left (c\,f-d\,e\right )}-\frac {2\,\left (e+f\,x\right )\,\left (a^3\,d^2\,f^3-3\,c\,a^2\,b\,d\,f^3-3\,a\,b^2\,d^2\,e^2\,f+6\,c\,a\,b^2\,d\,e\,f^2+2\,b^3\,d^2\,e^3-3\,c\,b^3\,d\,e^2\,f\right )}{{\left (c\,f-d\,e\right )}^2}}{d\,f^3\,{\left (e+f\,x\right )}^{3/2}}+\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\left (c^2\,d\,f^2-2\,c\,d^2\,e\,f+d^3\,e^2\right )}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{5/2}\,\left (2\,a^3\,d^3-6\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{d^{3/2}\,{\left (c\,f-d\,e\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/((e + f*x)^(5/2)*(c + d*x)),x)

[Out]

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

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