3.18.52 \(\int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^2} \, dx\) [1752]

Optimal. Leaf size=174 \[ \frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}-\frac {\sqrt {b d-a e} (2 b B d+3 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}} \]

[Out]

1/3*(3*A*b*e-5*B*a*e+2*B*b*d)*(e*x+d)^(3/2)/b^2/(-a*e+b*d)-(A*b-B*a)*(e*x+d)^(5/2)/b/(-a*e+b*d)/(b*x+a)-(3*A*b
*e-5*B*a*e+2*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(7/2)+(3*A*b*e-5*B*a*e+
2*B*b*d)*(e*x+d)^(1/2)/b^3

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Rubi [A]
time = 0.09, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 52, 65, 214} \begin {gather*} -\frac {\sqrt {b d-a e} (-5 a B e+3 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}}+\frac {\sqrt {d+e x} (-5 a B e+3 A b e+2 b B d)}{b^3}+\frac {(d+e x)^{3/2} (-5 a B e+3 A b e+2 b B d)}{3 b^2 (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{b (a+b x) (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(3/2))/(a + b*x)^2,x]

[Out]

((2*b*B*d + 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/b^3 + ((2*b*B*d + 3*A*b*e - 5*a*B*e)*(d + e*x)^(3/2))/(3*b^2*(b*
d - a*e)) - ((A*b - a*B)*(d + e*x)^(5/2))/(b*(b*d - a*e)*(a + b*x)) - (Sqrt[b*d - a*e]*(2*b*B*d + 3*A*b*e - 5*
a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(7/2)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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) (d+e x)^{3/2}}{(a+b x)^2} \, dx &=-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+3 A b e-5 a B e) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b (b d-a e)}\\ &=\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+3 A b e-5 a B e) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^2}\\ &=\frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {((b d-a e) (2 b B d+3 A b e-5 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^3}\\ &=\frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}+\frac {((b d-a e) (2 b B d+3 A b e-5 a B e)) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^3 e}\\ &=\frac {(2 b B d+3 A b e-5 a B e) \sqrt {d+e x}}{b^3}+\frac {(2 b B d+3 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{5/2}}{b (b d-a e) (a+b x)}-\frac {\sqrt {b d-a e} (2 b B d+3 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 139, normalized size = 0.80 \begin {gather*} \frac {\sqrt {d+e x} \left (3 A b (-b d+3 a e+2 b e x)+B \left (-15 a^2 e+a b (11 d-10 e x)+2 b^2 x (4 d+e x)\right )\right )}{3 b^3 (a+b x)}-\frac {\sqrt {-b d+a e} (2 b B d+3 A b e-5 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(3/2))/(a + b*x)^2,x]

[Out]

(Sqrt[d + e*x]*(3*A*b*(-(b*d) + 3*a*e + 2*b*e*x) + B*(-15*a^2*e + a*b*(11*d - 10*e*x) + 2*b^2*x*(4*d + e*x))))
/(3*b^3*(a + b*x)) - (Sqrt[-(b*d) + a*e]*(2*b*B*d + 3*A*b*e - 5*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b
*d) + a*e]])/b^(7/2)

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Maple [A]
time = 0.12, size = 195, normalized size = 1.12

method	result	size
derivativedivides	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A b e \sqrt {e x +d}-4 a e B \sqrt {e x +d}+2 B b d \sqrt {e x +d}}{b^{3}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A a b \,e^{2}+\frac {1}{2} A \,b^{2} d e +\frac {1}{2} B \,a^{2} e^{2}-\frac {1}{2} B a b d e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (3 A a b \,e^{2}-3 A \,b^{2} d e -5 B \,a^{2} e^{2}+7 B a b d e -2 b^{2} B \,d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{3}}\)	\(195\)
default	\(\frac {\frac {2 B b \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A b e \sqrt {e x +d}-4 a e B \sqrt {e x +d}+2 B b d \sqrt {e x +d}}{b^{3}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A a b \,e^{2}+\frac {1}{2} A \,b^{2} d e +\frac {1}{2} B \,a^{2} e^{2}-\frac {1}{2} B a b d e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (3 A a b \,e^{2}-3 A \,b^{2} d e -5 B \,a^{2} e^{2}+7 B a b d e -2 b^{2} B \,d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{3}}\)	\(195\)
risch	\(\frac {2 \left (b B x e +3 A b e -6 B a e +4 B b d \right ) \sqrt {e x +d}}{3 b^{3}}+\frac {\sqrt {e x +d}\, A \,e^{2} a}{b^{2} \left (b e x +a e \right )}-\frac {\sqrt {e x +d}\, A e d}{b \left (b e x +a e \right )}-\frac {\sqrt {e x +d}\, B \,a^{2} e^{2}}{b^{3} \left (b e x +a e \right )}+\frac {\sqrt {e x +d}\, B a e d}{b^{2} \left (b e x +a e \right )}-\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,e^{2} a}{b^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A e d}{b \sqrt {\left (a e -b d \right ) b}}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{2} e^{2}}{b^{3} \sqrt {\left (a e -b d \right ) b}}-\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B a e d}{b^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,d^{2}}{b \sqrt {\left (a e -b d \right ) b}}\)	\(358\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/b^3*(1/3*B*b*(e*x+d)^(3/2)+A*b*e*(e*x+d)^(1/2)-2*a*e*B*(e*x+d)^(1/2)+B*b*d*(e*x+d)^(1/2))-2/b^3*((-1/2*A*a*b
*e^2+1/2*A*b^2*d*e+1/2*B*a^2*e^2-1/2*B*a*b*d*e)*(e*x+d)^(1/2)/(b*(e*x+d)+a*e-b*d)+1/2*(3*A*a*b*e^2-3*A*b^2*d*e
-5*B*a^2*e^2+7*B*a*b*d*e-2*B*b^2*d^2)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(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)*(e*x+d)^(3/2)/(b*x+a)^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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 1.18, size = 391, normalized size = 2.25 \begin {gather*} \left [\frac {3 \, {\left (2 \, B b^{2} d x + 2 \, B a b d - {\left (5 \, B a^{2} - 3 \, A a b + {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} e\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) + 2 \, {\left (8 \, B b^{2} d x + {\left (11 \, B a b - 3 \, A b^{2}\right )} d + {\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \, {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} e\right )} \sqrt {x e + d}}{6 \, {\left (b^{4} x + a b^{3}\right )}}, -\frac {3 \, {\left (2 \, B b^{2} d x + 2 \, B a b d - {\left (5 \, B a^{2} - 3 \, A a b + {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} e\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (8 \, B b^{2} d x + {\left (11 \, B a b - 3 \, A b^{2}\right )} d + {\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \, {\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (b^{4} x + a b^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(2*B*b^2*d*x + 2*B*a*b*d - (5*B*a^2 - 3*A*a*b + (5*B*a*b - 3*A*b^2)*x)*e)*sqrt((b*d - a*e)/b)*log((2*b
*d - 2*sqrt(x*e + d)*b*sqrt((b*d - a*e)/b) + (b*x - a)*e)/(b*x + a)) + 2*(8*B*b^2*d*x + (11*B*a*b - 3*A*b^2)*d
+ (2*B*b^2*x^2 - 15*B*a^2 + 9*A*a*b - 2*(5*B*a*b - 3*A*b^2)*x)*e)*sqrt(x*e + d))/(b^4*x + a*b^3), -1/3*(3*(2*
B*b^2*d*x + 2*B*a*b*d - (5*B*a^2 - 3*A*a*b + (5*B*a*b - 3*A*b^2)*x)*e)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e +
d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (8*B*b^2*d*x + (11*B*a*b - 3*A*b^2)*d + (2*B*b^2*x^2 - 15*B*a^2 + 9*
A*a*b - 2*(5*B*a*b - 3*A*b^2)*x)*e)*sqrt(x*e + d))/(b^4*x + a*b^3)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1992 vs. \(2 (162) = 324\).
time = 195.70, size = 1992, normalized size = 11.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(3/2)/(b*x+a)**2,x)

[Out]

2*A*a**2*e**3*sqrt(d + e*x)/(2*a**2*b**2*e**2 - 2*a*b**3*d*e + 2*a*b**3*e**2*x - 2*b**4*d*e*x) - A*a**2*e**3*s
qrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3))
- b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) + A*a**2*e**3*sqrt(-1/(b*(a*e - b*d)**3))*lo
g(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e -
b*d)**3)) + sqrt(d + e*x))/(2*b**2) - 4*A*a*d*e**2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2
*x - 2*b**3*d*e*x) + A*a*d*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b
*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b - A*a*d*e**2*sqrt(
-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**
2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b - 4*A*a*e**2*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b**3*s
qrt(a*e/b - d)) - A*d**2*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*
sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + A*d**2*e*sqrt(-1/(b*(
a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*
sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + 2*A*d**2*e*sqrt(d + e*x)/(2*a**2*e**2 - 2*a*b*d*e + 2*a*b*e**
2*x - 2*b**2*d*e*x) + 4*A*d*e*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b**2*sqrt(a*e/b - d)) + 2*A*e*sqrt(d + e*x)
/b**2 - 2*B*a**3*e**3*sqrt(d + e*x)/(2*a**2*b**3*e**2 - 2*a*b**4*d*e + 2*a*b**4*e**2*x - 2*b**5*d*e*x) + B*a**
3*e**3*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*
d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**3) - B*a**3*e**3*sqrt(-1/(b*(a*e - b*d)
**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b
*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**3) + 4*B*a**2*d*e**2*sqrt(d + e*x)/(2*a**2*b**2*e**2 - 2*a*b**3*d*e +
2*a*b**3*e**2*x - 2*b**4*d*e*x) - B*a**2*d*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e -
b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b**
2 + B*a**2*d*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b
*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b**2 + 6*B*a**2*e**2*atan(sqrt(d +
e*x)/sqrt(a*e/b - d))/(b**4*sqrt(a*e/b - d)) - 2*B*a*d**2*e*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*
b**2*e**2*x - 2*b**3*d*e*x) + B*a*d**2*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)
) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - B*a
*d**2*e*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*
d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - 8*B*a*d*e*atan(sqrt(d + e*x)/sqrt(a*e
/b - d))/(b**3*sqrt(a*e/b - d)) - 4*B*a*e*sqrt(d + e*x)/b**3 + 2*B*d**2*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b
**2*sqrt(a*e/b - d)) + 2*B*d*sqrt(d + e*x)/b**2 + 2*B*(d + e*x)**(3/2)/(3*b**2)

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Giac [A]
time = 0.89, size = 239, normalized size = 1.37 \begin {gather*} \frac {{\left (2 \, B b^{2} d^{2} - 7 \, B a b d e + 3 \, A b^{2} d e + 5 \, B a^{2} e^{2} - 3 \, A a b e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{3}} + \frac {\sqrt {x e + d} B a b d e - \sqrt {x e + d} A b^{2} d e - \sqrt {x e + d} B a^{2} e^{2} + \sqrt {x e + d} A a b e^{2}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{3}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{4} + 3 \, \sqrt {x e + d} B b^{4} d - 6 \, \sqrt {x e + d} B a b^{3} e + 3 \, \sqrt {x e + d} A b^{4} e\right )}}{3 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(b*x+a)^2,x, algorithm="giac")

[Out]

(2*B*b^2*d^2 - 7*B*a*b*d*e + 3*A*b^2*d*e + 5*B*a^2*e^2 - 3*A*a*b*e^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b
*e))/(sqrt(-b^2*d + a*b*e)*b^3) + (sqrt(x*e + d)*B*a*b*d*e - sqrt(x*e + d)*A*b^2*d*e - sqrt(x*e + d)*B*a^2*e^2
+ sqrt(x*e + d)*A*a*b*e^2)/(((x*e + d)*b - b*d + a*e)*b^3) + 2/3*((x*e + d)^(3/2)*B*b^4 + 3*sqrt(x*e + d)*B*b
^4*d - 6*sqrt(x*e + d)*B*a*b^3*e + 3*sqrt(x*e + d)*A*b^4*e)/b^6

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Mupad [B]
time = 1.32, size = 174, normalized size = 1.00 \begin {gather*} \left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (B\,a^2\,e^2-A\,a\,b\,e^2-B\,d\,a\,b\,e+A\,d\,b^2\,e\right )}{b^4\,\left (d+e\,x\right )-b^4\,d+a\,b^3\,e}+\frac {2\,B\,{\left (d+e\,x\right )}^{3/2}}{3\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (3\,A\,b\,e-5\,B\,a\,e+2\,B\,b\,d\right )\,1{}\mathrm {i}}{b^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(3/2))/(a + b*x)^2,x)

[Out]

((2*A*e - 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b*e))/b^4)*(d + e*x)^(1/2) - ((d + e*x)^(1/2)*(B*a^2*e^2 - A*a*b*e^
2 + A*b^2*d*e - B*a*b*d*e))/(b^4*(d + e*x) - b^4*d + a*b^3*e) + (2*B*(d + e*x)^(3/2))/(3*b^2) + (atan((b^(1/2)
*(d + e*x)^(1/2)*1i)/(b*d - a*e)^(1/2))*(b*d - a*e)^(1/2)*(3*A*b*e - 5*B*a*e + 2*B*b*d)*1i)/b^(7/2)

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