3.18.79 \(\int \frac {(a+b x)^2}{(c+d x) (e+f x)^{5/2}} \, dx\) [1779]

Optimal. Leaf size=140 \[ \frac {2 (b e-a f)^2}{3 f^2 (d e-c f) (e+f x)^{3/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{f^2 (d e-c f)^2 \sqrt {e+f x}}-\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 140, 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 e-a f) (a d f-2 b c f+b d e)}{f^2 \sqrt {e+f x} (d e-c f)^2}+\frac {2 (b e-a f)^2}{3 f^2 (e+f x)^{3/2} (d e-c f)}-\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.28, size = 135, normalized size = 0.96 \begin {gather*} -\frac {2 (b e-a f) (b d e (2 e+3 f x)-b c f (5 e+6 f x)+a f (4 d e-c f+3 d f x))}{3 f^2 (d e-c f)^2 (e+f x)^{3/2}}+\frac {2 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {d} (-d e+c f)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 169, normalized size = 1.21

method	result	size
derivativedivides	\(\frac {\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}-\frac {2 \left (-a^{2} d \,f^{2}+2 a b c \,f^{2}-2 b^{2} c e f +b^{2} d \,e^{2}\right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f^{2}}\)	\(169\)
default	\(\frac {\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}-\frac {2 \left (-a^{2} d \,f^{2}+2 a b c \,f^{2}-2 b^{2} c e f +b^{2} d \,e^{2}\right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f^{2}}\)	\(169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f^2*(f^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(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^2*f^2-2*a*b*e*f+b^2*e^2)/(c*f-d*e)/(f*x+e)^(3/2)-1/(c*f-d*e)^2*(-a^2*d*f^2+2*a*b*c*f^2-2*b^2*c*e*f
+b^2*d*e^2)/(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)^(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. 485 vs. \(2 (136) = 272\).
time = 0.79, size = 984, normalized size = 7.03 \begin {gather*} \left [-\frac {3 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{4} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x e + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} 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 (a^{2} c^{2} d f^{4} + 3 \, {\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{4} x - 2 \, b^{2} d^{3} e^{4} - {\left (3 \, b^{2} d^{3} f x - {\left (7 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} f\right )} e^{3} + {\left (9 \, b^{2} c d^{2} f^{2} x - {\left (5 \, b^{2} c^{2} d + 2 \, a b c d^{2} - 4 \, a^{2} d^{3}\right )} f^{2}\right )} e^{2} - {\left (3 \, {\left (2 \, b^{2} c^{2} d + 2 \, a b c d^{2} - a^{2} d^{3}\right )} f^{3} x - {\left (4 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} f^{3}\right )} e\right )} \sqrt {f x + e}}{3 \, {\left (c^{3} d f^{7} x^{2} - d^{4} f^{2} e^{5} - {\left (2 \, d^{4} f^{3} x - 3 \, c d^{3} f^{3}\right )} e^{4} - {\left (d^{4} f^{4} x^{2} - 6 \, c d^{3} f^{4} x + 3 \, c^{2} d^{2} f^{4}\right )} e^{3} + {\left (3 \, c d^{3} f^{5} x^{2} - 6 \, c^{2} d^{2} f^{5} x + c^{3} d f^{5}\right )} e^{2} - {\left (3 \, c^{2} d^{2} f^{6} x^{2} - 2 \, c^{3} d f^{6} x\right )} e\right )}}, -\frac {2 \, {\left (3 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{4} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x e + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} 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 (a^{2} c^{2} d f^{4} + 3 \, {\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{4} x - 2 \, b^{2} d^{3} e^{4} - {\left (3 \, b^{2} d^{3} f x - {\left (7 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} f\right )} e^{3} + {\left (9 \, b^{2} c d^{2} f^{2} x - {\left (5 \, b^{2} c^{2} d + 2 \, a b c d^{2} - 4 \, a^{2} d^{3}\right )} f^{2}\right )} e^{2} - {\left (3 \, {\left (2 \, b^{2} c^{2} d + 2 \, a b c d^{2} - a^{2} d^{3}\right )} f^{3} x - {\left (4 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} f^{3}\right )} e\right )} \sqrt {f x + e}\right )}}{3 \, {\left (c^{3} d f^{7} x^{2} - d^{4} f^{2} e^{5} - {\left (2 \, d^{4} f^{3} x - 3 \, c d^{3} f^{3}\right )} e^{4} - {\left (d^{4} f^{4} x^{2} - 6 \, c d^{3} f^{4} x + 3 \, c^{2} d^{2} f^{4}\right )} e^{3} + {\left (3 \, c d^{3} f^{5} x^{2} - 6 \, c^{2} d^{2} f^{5} x + c^{3} d f^{5}\right )} e^{2} - {\left (3 \, c^{2} d^{2} f^{6} x^{2} - 2 \, c^{3} d f^{6} x\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)^(5/2),x, algorithm="fricas")

[Out]

[-1/3*(3*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^4*x^2 + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^3*x*e + (b^2*c^2 - 2*a
*b*c*d + a^2*d^2)*f^2*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*(a^2*c^2*d*f^4 + 3*(2*a*b*c^2*d - a^2*c*d^2)*f^4*x - 2*b^2*d^3*e^4 - (3*b^2*d^3*f*x - (7*b^2
*c*d^2 - 2*a*b*d^3)*f)*e^3 + (9*b^2*c*d^2*f^2*x - (5*b^2*c^2*d + 2*a*b*c*d^2 - 4*a^2*d^3)*f^2)*e^2 - (3*(2*b^2
*c^2*d + 2*a*b*c*d^2 - a^2*d^3)*f^3*x - (4*a*b*c^2*d - 5*a^2*c*d^2)*f^3)*e)*sqrt(f*x + e))/(c^3*d*f^7*x^2 - d^
4*f^2*e^5 - (2*d^4*f^3*x - 3*c*d^3*f^3)*e^4 - (d^4*f^4*x^2 - 6*c*d^3*f^4*x + 3*c^2*d^2*f^4)*e^3 + (3*c*d^3*f^5
*x^2 - 6*c^2*d^2*f^5*x + c^3*d*f^5)*e^2 - (3*c^2*d^2*f^6*x^2 - 2*c^3*d*f^6*x)*e), -2/3*(3*((b^2*c^2 - 2*a*b*c*
d + a^2*d^2)*f^4*x^2 + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^3*x*e + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*e^2)*sq
rt(c*d*f - d^2*e)*arctan(sqrt(c*d*f - d^2*e)*sqrt(f*x + e)/(d*f*x + d*e)) + (a^2*c^2*d*f^4 + 3*(2*a*b*c^2*d -
a^2*c*d^2)*f^4*x - 2*b^2*d^3*e^4 - (3*b^2*d^3*f*x - (7*b^2*c*d^2 - 2*a*b*d^3)*f)*e^3 + (9*b^2*c*d^2*f^2*x - (5
*b^2*c^2*d + 2*a*b*c*d^2 - 4*a^2*d^3)*f^2)*e^2 - (3*(2*b^2*c^2*d + 2*a*b*c*d^2 - a^2*d^3)*f^3*x - (4*a*b*c^2*d
- 5*a^2*c*d^2)*f^3)*e)*sqrt(f*x + e))/(c^3*d*f^7*x^2 - d^4*f^2*e^5 - (2*d^4*f^3*x - 3*c*d^3*f^3)*e^4 - (d^4*f
^4*x^2 - 6*c*d^3*f^4*x + 3*c^2*d^2*f^4)*e^3 + (3*c*d^3*f^5*x^2 - 6*c^2*d^2*f^5*x + c^3*d*f^5)*e^2 - (3*c^2*d^2
*f^6*x^2 - 2*c^3*d*f^6*x)*e)]

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Sympy [A]
time = 66.33, size = 129, normalized size = 0.92 \begin {gather*} \frac {2 \left (a f - b e\right ) \left (a d f - 2 b c f + b d e\right )}{f^{2} \sqrt {e + f x} \left (c f - d e\right )^{2}} - \frac {2 \left (a f - b e\right )^{2}}{3 f^{2} \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )} + \frac {2 \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d \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)**2/(d*x+c)/(f*x+e)**(5/2),x)

[Out]

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

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Giac [A]
time = 0.79, size = 236, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{{\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \sqrt {c d f - d^{2} e}} - \frac {2 \, {\left (6 \, {\left (f x + e\right )} a b c f^{2} - 3 \, {\left (f x + e\right )} a^{2} d f^{2} + a^{2} c f^{3} - 6 \, {\left (f x + e\right )} b^{2} c f e - 2 \, a b c f^{2} e - a^{2} d f^{2} e + 3 \, {\left (f x + e\right )} b^{2} d e^{2} + b^{2} c f e^{2} + 2 \, a b d f e^{2} - b^{2} d e^{3}\right )}}{3 \, {\left (c^{2} f^{4} - 2 \, c d f^{3} e + d^{2} f^{2} 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)^2/(d*x+c)/(f*x+e)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.36, size = 203, normalized size = 1.45 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2\right )}{{\left (c\,f-d\,e\right )}^{5/2}\,\left (2\,a^2\,d^2-4\,a\,b\,c\,d+2\,b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{5/2}}-\frac {\frac {2\,\left (a^2\,f^2-2\,a\,b\,e\,f+b^2\,e^2\right )}{3\,\left (c\,f-d\,e\right )}-\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 )}{{\left (c\,f-d\,e\right )}^2}}{f^2\,{\left (e+f\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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