\(\int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx\) [2255]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=\frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\sqrt {e} (3 b B d+2 A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 49, 52, 65, 223, 212} \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {e} (-5 a B e+2 A b e+3 b B d) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}}+\frac {e \sqrt {a+b x} \sqrt {d+e x} (-5 a B e+2 A b e+3 b B d)}{b^3 (b d-a e)}-\frac {2 (d+e x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{3 b^2 \sqrt {a+b x} (b d-a e)}-\frac {2 (d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

[In]

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

[Out]

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

Rule 49

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

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(3 b B d+2 A b e-5 a B e) \int \frac {(d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx}{3 b (b d-a e)} \\ & = -\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a B e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{b^2 (b d-a e)} \\ & = \frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{2 b^3} \\ & = \frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a B e)) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^4} \\ & = \frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a B e)) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b^4} \\ & = \frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\sqrt {e} (3 b B d+2 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.66 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=-\frac {\sqrt {d+e x} \left (B \left (-15 a^2 e+4 a b (d-5 e x)-3 b^2 x (-2 d+e x)\right )+2 A b (3 a e+b (d+4 e x))\right )}{3 b^3 (a+b x)^{3/2}}+\frac {\sqrt {e} (3 b B d+2 A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}} \]

[In]

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

[Out]

-1/3*(Sqrt[d + e*x]*(B*(-15*a^2*e + 4*a*b*(d - 5*e*x) - 3*b^2*x*(-2*d + e*x)) + 2*A*b*(3*a*e + b*(d + 4*e*x)))
)/(b^3*(a + b*x)^(3/2)) + (Sqrt[e]*(3*b*B*d + 2*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt
[d + e*x])])/b^(7/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(697\) vs. \(2(173)=346\).

Time = 1.11 (sec) , antiderivative size = 698, normalized size of antiderivative = 3.47

method	result	size
default	\(\frac {\sqrt {e x +d}\, \left (6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} e^{2} x^{2}-15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} e^{2} x^{2}+9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d e \,x^{2}+12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} e^{2} x -30 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b \,e^{2} x +18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d e x +6 B \,b^{2} e \,x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b \,e^{2}-16 A \,b^{2} e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} e^{2}+9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b d e +40 B a b e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-12 B \,b^{2} d x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-12 A a b e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-4 A \,b^{2} d \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+30 B \,a^{2} e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-8 B a b d \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{6 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} \left (b x +a \right )^{\frac {3}{2}}}\)	\(698\)

[In]

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

[Out]

1/6*(e*x+d)^(1/2)*(6*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*e^2*x^2
-15*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*e^2*x^2+9*B*ln(1/2*(2*
b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d*e*x^2+12*A*ln(1/2*(2*b*e*x+2*((b*x+a)*
(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*e^2*x-30*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*e^2*x+18*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d
)/(b*e)^(1/2))*a*b^2*d*e*x+6*B*b^2*e*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+6*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*e^2-16*A*b^2*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-15*B
*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*e^2+9*B*ln(1/2*(2*b*e*x+2*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*d*e+40*B*a*b*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1
/2)-12*B*b^2*d*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-12*A*a*b*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-4*A*b^2*d*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+30*B*a^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-8*B*a*b*d*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2))/(b*e)^(1/2)/((b*x+a)*(e*x+d))^(1/2)/b^3/(b*x+a)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.92 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.79 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=\left [\frac {3 \, {\left (3 \, B a^{2} b d + {\left (3 \, B b^{3} d - {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{2} - {\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} e + 2 \, {\left (3 \, B a b^{2} d - {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} e\right )} x\right )} \sqrt {\frac {e}{b}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b^{2} e x + b^{2} d + a b e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {e}{b}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (3 \, B b^{2} e x^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d + 3 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} e - 2 \, {\left (3 \, B b^{2} d - 2 \, {\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{12 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {3 \, {\left (3 \, B a^{2} b d + {\left (3 \, B b^{3} d - {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{2} - {\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} e + 2 \, {\left (3 \, B a b^{2} d - {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} e\right )} x\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {e}{b}}}{2 \, {\left (b e^{2} x^{2} + a d e + {\left (b d e + a e^{2}\right )} x\right )}}\right ) - 2 \, {\left (3 \, B b^{2} e x^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d + 3 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} e - 2 \, {\left (3 \, B b^{2} d - 2 \, {\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{6 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]

[In]

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

[Out]

[1/12*(3*(3*B*a^2*b*d + (3*B*b^3*d - (5*B*a*b^2 - 2*A*b^3)*e)*x^2 - (5*B*a^3 - 2*A*a^2*b)*e + 2*(3*B*a*b^2*d -
(5*B*a^2*b - 2*A*a*b^2)*e)*x)*sqrt(e/b)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b^2*e*x + b^
2*d + a*b*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(3*B*b^2*e*x^2 - 2*(2*B*a*b
+ A*b^2)*d + 3*(5*B*a^2 - 2*A*a*b)*e - 2*(3*B*b^2*d - 2*(5*B*a*b - 2*A*b^2)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d))
/(b^5*x^2 + 2*a*b^4*x + a^2*b^3), -1/6*(3*(3*B*a^2*b*d + (3*B*b^3*d - (5*B*a*b^2 - 2*A*b^3)*e)*x^2 - (5*B*a^3
- 2*A*a^2*b)*e + 2*(3*B*a*b^2*d - (5*B*a^2*b - 2*A*a*b^2)*e)*x)*sqrt(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)*sq
rt(b*x + a)*sqrt(e*x + d)*sqrt(-e/b)/(b*e^2*x^2 + a*d*e + (b*d*e + a*e^2)*x)) - 2*(3*B*b^2*e*x^2 - 2*(2*B*a*b
+ A*b^2)*d + 3*(5*B*a^2 - 2*A*a*b)*e - 2*(3*B*b^2*d - 2*(5*B*a*b - 2*A*b^2)*e)*x)*sqrt(b*x + a)*sqrt(e*x + d))
/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)]

Sympy [F]

\[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((B*x+A)*(e*x+d)**(3/2)/(b*x+a)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**(3/2)/(a + b*x)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(b*x+a)^(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(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (173) = 346\).

Time = 0.53 (sec) , antiderivative size = 1019, normalized size of antiderivative = 5.07 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} B e {\left | b \right |}}{b^{5}} - \frac {{\left (3 \, \sqrt {b e} B b d {\left | b \right |} - 5 \, \sqrt {b e} B a e {\left | b \right |} + 2 \, \sqrt {b e} A b e {\left | b \right |}\right )} \log \left ({\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{2 \, b^{5}} - \frac {4 \, {\left (3 \, \sqrt {b e} B b^{6} d^{4} {\left | b \right |} - 16 \, \sqrt {b e} B a b^{5} d^{3} e {\left | b \right |} + 4 \, \sqrt {b e} A b^{6} d^{3} e {\left | b \right |} + 30 \, \sqrt {b e} B a^{2} b^{4} d^{2} e^{2} {\left | b \right |} - 12 \, \sqrt {b e} A a b^{5} d^{2} e^{2} {\left | b \right |} - 24 \, \sqrt {b e} B a^{3} b^{3} d e^{3} {\left | b \right |} + 12 \, \sqrt {b e} A a^{2} b^{4} d e^{3} {\left | b \right |} + 7 \, \sqrt {b e} B a^{4} b^{2} e^{4} {\left | b \right |} - 4 \, \sqrt {b e} A a^{3} b^{3} e^{4} {\left | b \right |} - 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{4} d^{3} {\left | b \right |} + 24 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a b^{3} d^{2} e {\left | b \right |} - 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{4} d^{2} e {\left | b \right |} - 30 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{2} b^{2} d e^{2} {\left | b \right |} + 12 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A a b^{3} d e^{2} {\left | b \right |} + 12 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{3} b e^{3} {\left | b \right |} - 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A a^{2} b^{2} e^{3} {\left | b \right |} + 3 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B b^{2} d^{2} {\left | b \right |} - 12 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B a b d e {\left | b \right |} + 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} A b^{2} d e {\left | b \right |} + 9 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B a^{2} e^{2} {\left | b \right |} - 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} A a b e^{2} {\left | b \right |}\right )}}{3 \, {\left (b^{2} d - a b e - {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3} b^{4}} \]

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*B*e*abs(b)/b^5 - 1/2*(3*sqrt(b*e)*B*b*d*abs(b) - 5*sqrt(b*e)
*B*a*e*abs(b) + 2*sqrt(b*e)*A*b*e*abs(b))*log((sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^
2)/b^5 - 4/3*(3*sqrt(b*e)*B*b^6*d^4*abs(b) - 16*sqrt(b*e)*B*a*b^5*d^3*e*abs(b) + 4*sqrt(b*e)*A*b^6*d^3*e*abs(b
) + 30*sqrt(b*e)*B*a^2*b^4*d^2*e^2*abs(b) - 12*sqrt(b*e)*A*a*b^5*d^2*e^2*abs(b) - 24*sqrt(b*e)*B*a^3*b^3*d*e^3
*abs(b) + 12*sqrt(b*e)*A*a^2*b^4*d*e^3*abs(b) + 7*sqrt(b*e)*B*a^4*b^2*e^4*abs(b) - 4*sqrt(b*e)*A*a^3*b^3*e^4*a
bs(b) - 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^4*d^3*abs(b) + 24*sq
rt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^3*d^2*e*abs(b) - 6*sqrt(b*e)*(
sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^4*d^2*e*abs(b) - 30*sqrt(b*e)*(sqrt(b*e)*
sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^2*b^2*d*e^2*abs(b) + 12*sqrt(b*e)*(sqrt(b*e)*sqrt(b
*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a*b^3*d*e^2*abs(b) + 12*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a)
- sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^3*b*e^3*abs(b) - 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*
d + (b*x + a)*b*e - a*b*e))^2*A*a^2*b^2*e^3*abs(b) + 3*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e))^4*B*b^2*d^2*abs(b) - 12*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a
*b*e))^4*B*a*b*d*e*abs(b) + 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^
2*d*e*abs(b) + 9*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^2*e^2*abs(b)
- 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*a*b*e^2*abs(b))/((b^2*d - a*
b*e - (sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)^3*b^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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