3.23.45 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx\) [2245]

Optimal. Leaf size=249 \[ \frac {5 (b d-a e) (b B d+6 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {5 (b d-a e)^2 (b B d+6 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \]

[Out]

5/8*(-a*e+b*d)^2*(6*A*b*e-7*B*a*e+B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(9/2)/e^(1/2)-
2*(A*b-B*a)*(e*x+d)^(7/2)/b/(-a*e+b*d)/(b*x+a)^(1/2)+5/12*(6*A*b*e-7*B*a*e+B*b*d)*(e*x+d)^(3/2)*(b*x+a)^(1/2)/
b^3+1/3*(6*A*b*e-7*B*a*e+B*b*d)*(e*x+d)^(5/2)*(b*x+a)^(1/2)/b^2/(-a*e+b*d)+5/8*(-a*e+b*d)*(6*A*b*e-7*B*a*e+B*b
*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^4

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Rubi [A]
time = 0.13, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 52, 65, 223, 212} \begin {gather*} \frac {5 (b d-a e)^2 (-7 a B e+6 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}}+\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e) (-7 a B e+6 A b e+b B d)}{8 b^4}+\frac {5 \sqrt {a+b x} (d+e x)^{3/2} (-7 a B e+6 A b e+b B d)}{12 b^3}+\frac {\sqrt {a+b x} (d+e x)^{5/2} (-7 a B e+6 A b e+b B d)}{3 b^2 (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b*d - a*e)*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^4) + (5*(b*B*d + 6*A*b*e - 7*a*B*
e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^3) + ((b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^
2*(b*d - a*e)) - (2*(A*b - a*B)*(d + e*x)^(7/2))/(b*(b*d - a*e)*Sqrt[a + b*x]) + (5*(b*d - a*e)^2*(b*B*d + 6*A
*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(9/2)*Sqrt[e])

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*} \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx &=-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(b B d+6 A b e-7 a B e) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}} \, dx}{b (b d-a e)}\\ &=\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(5 (b B d+6 A b e-7 a B e)) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^2}\\ &=\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(5 (b d-a e) (b B d+6 A b e-7 a B e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{8 b^3}\\ &=\frac {5 (b d-a e) (b B d+6 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {\left (5 (b d-a e)^2 (b B d+6 A b e-7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{16 b^4}\\ &=\frac {5 (b d-a e) (b B d+6 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {\left (5 (b d-a e)^2 (b B d+6 A b e-7 a B e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^5}\\ &=\frac {5 (b d-a e) (b B d+6 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {\left (5 (b d-a e)^2 (b B d+6 A b e-7 a B e)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{8 b^5}\\ &=\frac {5 (b d-a e) (b B d+6 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{7/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {5 (b d-a e)^2 (b B d+6 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}}\\ \end {align*}

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Mathematica [A]
time = 0.68, size = 218, normalized size = 0.88 \begin {gather*} \frac {\sqrt {d+e x} \left (-6 A b \left (15 a^2 e^2+5 a b e (-5 d+e x)+b^2 \left (8 d^2-9 d e x-2 e^2 x^2\right )\right )+B \left (105 a^3 e^2+5 a^2 b e (-38 d+7 e x)+a b^2 \left (81 d^2-68 d e x-14 e^2 x^2\right )+b^3 x \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )\right )}{24 b^4 \sqrt {a+b x}}+\frac {5 (b d-a e)^2 (b B d+6 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{9/2} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-6*A*b*(15*a^2*e^2 + 5*a*b*e*(-5*d + e*x) + b^2*(8*d^2 - 9*d*e*x - 2*e^2*x^2)) + B*(105*a^3*e^
2 + 5*a^2*b*e*(-38*d + 7*e*x) + a*b^2*(81*d^2 - 68*d*e*x - 14*e^2*x^2) + b^3*x*(33*d^2 + 26*d*e*x + 8*e^2*x^2)
)))/(24*b^4*Sqrt[a + b*x]) + (5*(b*d - a*e)^2*(b*B*d + 6*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqr
t[e]*Sqrt[a + b*x])])/(8*b^(9/2)*Sqrt[e])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1183\) vs. \(2(213)=426\).
time = 0.11, size = 1184, normalized size = 4.76

method	result	size
default	\(\frac {\sqrt {e x +d}\, \left (-105 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} b \,e^{3} x -180 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^{2} d \,e^{2}+90 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^{3} d^{2} e +225 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} b d \,e^{2}-135 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^{2} d^{2} e +16 B \,b^{3} e^{2} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+24 A \,b^{3} e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+66 B \,b^{3} d^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-180 A \,a^{2} b \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+162 B a \,b^{2} d^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+90 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^{2} e^{3} x +90 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^{4} d^{2} e x -105 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^{4} e^{3}-136 B a \,b^{2} d e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-180 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^{3} d \,e^{2} x +225 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^{2} d \,e^{2} x -96 A \,b^{3} d^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+210 B \,a^{3} e^{2} \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 ) b^{4} d^{3} x +90 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^{3} b \,e^{3}+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^{3} d^{3}-135 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^{3} d^{2} e x -28 B a \,b^{2} e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+52 B \,b^{3} d e \,x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-60 A a \,b^{2} e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+108 A \,b^{3} d e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+70 B \,a^{2} b \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+300 A a \,b^{2} d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-380 B \,a^{2} b d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{48 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, \sqrt {b x +a}\, b^{4}}\)	\(1184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(e*x+d)^(1/2)*(-105*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*b*e
^3*x-180*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^2*d*e^2+90*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^3*d^2*e+225*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*b*d*e^2-135*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^2*d^2*e+16*B*b^3*e^2*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+24*
A*b^3*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+66*B*b^3*d^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*A*a^2
*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+162*B*a*b^2*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*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^2*e^3*x+90*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^4*d^2*e*x-105*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^4*e^3-136*B*a*b^2*d*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*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^3*d*e^2*x+225*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^2*d*e^2*x-96*A*b^3*d^2*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)+210*B*a^3*e^2*((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))*b^4*d^3*x+90*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^3*b*e^3+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^3*d
^3-135*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^3*d^2*e*x-28*B*a*b^2*
e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+52*B*b^3*d*e*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-60*A*a*b^2*e^
2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+108*A*b^3*d*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+70*B*a^2*b*e^2*x*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+300*A*a*b^2*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-380*B*a^2*b*d*e*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(1/2)/b^4

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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)^(5/2)/(b*x+a)^(3/2),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 = 1.23, size = 838, normalized size = 3.37 \begin {gather*} \left [-\frac {{\left (15 \, {\left (B b^{4} d^{3} x + B a b^{3} d^{3} - {\left (7 \, B a^{4} - 6 \, A a^{3} b + {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} e^{3} + 3 \, {\left ({\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d x + {\left (5 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} d\right )} e^{2} - 3 \, {\left ({\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d^{2} x + {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} - 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) - 4 \, \sqrt {b x + a} {\left ({\left (8 \, B b^{4} x^{3} + 105 \, B a^{3} b - 90 \, A a^{2} b^{2} - 2 \, {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x\right )} e^{3} + 2 \, {\left (13 \, B b^{4} d x^{2} - {\left (34 \, B a b^{3} - 27 \, A b^{4}\right )} d x - 5 \, {\left (19 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d\right )} e^{2} + 3 \, {\left (11 \, B b^{4} d^{2} x + {\left (27 \, B a b^{3} - 16 \, A b^{4}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{96 \, {\left (b^{6} x + a b^{5}\right )}}, -\frac {{\left (15 \, {\left (B b^{4} d^{3} x + B a b^{3} d^{3} - {\left (7 \, B a^{4} - 6 \, A a^{3} b + {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} e^{3} + 3 \, {\left ({\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d x + {\left (5 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} d\right )} e^{2} - 3 \, {\left ({\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d^{2} x + {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) - 2 \, \sqrt {b x + a} {\left ({\left (8 \, B b^{4} x^{3} + 105 \, B a^{3} b - 90 \, A a^{2} b^{2} - 2 \, {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x\right )} e^{3} + 2 \, {\left (13 \, B b^{4} d x^{2} - {\left (34 \, B a b^{3} - 27 \, A b^{4}\right )} d x - 5 \, {\left (19 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d\right )} e^{2} + 3 \, {\left (11 \, B b^{4} d^{2} x + {\left (27 \, B a b^{3} - 16 \, A b^{4}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{48 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(15*(B*b^4*d^3*x + B*a*b^3*d^3 - (7*B*a^4 - 6*A*a^3*b + (7*B*a^3*b - 6*A*a^2*b^2)*x)*e^3 + 3*((5*B*a^2*
b^2 - 4*A*a*b^3)*d*x + (5*B*a^3*b - 4*A*a^2*b^2)*d)*e^2 - 3*((3*B*a*b^3 - 2*A*b^4)*d^2*x + (3*B*a^2*b^2 - 2*A*
a*b^3)*d^2)*e)*sqrt(b)*e^(1/2)*log(b^2*d^2 - 4*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(b)*e^(1/
2) + (8*b^2*x^2 + 8*a*b*x + a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e) - 4*sqrt(b*x + a)*((8*B*b^4*x^3 + 105*B*a^3*
b - 90*A*a^2*b^2 - 2*(7*B*a*b^3 - 6*A*b^4)*x^2 + 5*(7*B*a^2*b^2 - 6*A*a*b^3)*x)*e^3 + 2*(13*B*b^4*d*x^2 - (34*
B*a*b^3 - 27*A*b^4)*d*x - 5*(19*B*a^2*b^2 - 15*A*a*b^3)*d)*e^2 + 3*(11*B*b^4*d^2*x + (27*B*a*b^3 - 16*A*b^4)*d
^2)*e)*sqrt(x*e + d))*e^(-1)/(b^6*x + a*b^5), -1/48*(15*(B*b^4*d^3*x + B*a*b^3*d^3 - (7*B*a^4 - 6*A*a^3*b + (7
*B*a^3*b - 6*A*a^2*b^2)*x)*e^3 + 3*((5*B*a^2*b^2 - 4*A*a*b^3)*d*x + (5*B*a^3*b - 4*A*a^2*b^2)*d)*e^2 - 3*((3*B
*a*b^3 - 2*A*b^4)*d^2*x + (3*B*a^2*b^2 - 2*A*a*b^3)*d^2)*e)*sqrt(-b*e)*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(b
*x + a)*sqrt(-b*e)*sqrt(x*e + d)/((b^2*x^2 + a*b*x)*e^2 + (b^2*d*x + a*b*d)*e)) - 2*sqrt(b*x + a)*((8*B*b^4*x^
3 + 105*B*a^3*b - 90*A*a^2*b^2 - 2*(7*B*a*b^3 - 6*A*b^4)*x^2 + 5*(7*B*a^2*b^2 - 6*A*a*b^3)*x)*e^3 + 2*(13*B*b^
4*d*x^2 - (34*B*a*b^3 - 27*A*b^4)*d*x - 5*(19*B*a^2*b^2 - 15*A*a*b^3)*d)*e^2 + 3*(11*B*b^4*d^2*x + (27*B*a*b^3
- 16*A*b^4)*d^2)*e)*sqrt(x*e + d))*e^(-1)/(b^6*x + a*b^5)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (228) = 456\).
time = 0.86, size = 477, normalized size = 1.92 \begin {gather*} \frac {1}{24} \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |} e^{2}}{b^{6}} + \frac {{\left (13 \, B b^{18} d {\left | b \right |} e^{5} - 19 \, B a b^{17} {\left | b \right |} e^{6} + 6 \, A b^{18} {\left | b \right |} e^{6}\right )} e^{\left (-4\right )}}{b^{23}}\right )} + \frac {3 \, {\left (11 \, B b^{19} d^{2} {\left | b \right |} e^{4} - 40 \, B a b^{18} d {\left | b \right |} e^{5} + 18 \, A b^{19} d {\left | b \right |} e^{5} + 29 \, B a^{2} b^{17} {\left | b \right |} e^{6} - 18 \, A a b^{18} {\left | b \right |} e^{6}\right )} e^{\left (-4\right )}}{b^{23}}\right )} - \frac {5 \, {\left (B b^{\frac {7}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - 9 \, B a b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 6 \, A b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 15 \, B a^{2} b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 12 \, A a b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 7 \, B a^{3} \sqrt {b} {\left | b \right |} e^{\frac {7}{2}} + 6 \, A a^{2} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {7}{2}}\right )} e^{\left (-1\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{16 \, b^{6}} + \frac {4 \, {\left (B a b^{\frac {7}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - A b^{\frac {9}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - 3 \, B a^{2} b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 3 \, A a b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 3 \, B a^{3} b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 3 \, A a^{2} b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {5}{2}} - B a^{4} \sqrt {b} {\left | b \right |} e^{\frac {7}{2}} + A a^{3} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {7}{2}}\right )}}{{\left (b^{2} d - a b e - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*B*abs(b)*e^2/b^6 + (13*B*b^18
*d*abs(b)*e^5 - 19*B*a*b^17*abs(b)*e^6 + 6*A*b^18*abs(b)*e^6)*e^(-4)/b^23) + 3*(11*B*b^19*d^2*abs(b)*e^4 - 40*
B*a*b^18*d*abs(b)*e^5 + 18*A*b^19*d*abs(b)*e^5 + 29*B*a^2*b^17*abs(b)*e^6 - 18*A*a*b^18*abs(b)*e^6)*e^(-4)/b^2
3) - 5/16*(B*b^(7/2)*d^3*abs(b)*e^(1/2) - 9*B*a*b^(5/2)*d^2*abs(b)*e^(3/2) + 6*A*b^(7/2)*d^2*abs(b)*e^(3/2) +
15*B*a^2*b^(3/2)*d*abs(b)*e^(5/2) - 12*A*a*b^(5/2)*d*abs(b)*e^(5/2) - 7*B*a^3*sqrt(b)*abs(b)*e^(7/2) + 6*A*a^2
*b^(3/2)*abs(b)*e^(7/2))*e^(-1)*log((sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/b
^6 + 4*(B*a*b^(7/2)*d^3*abs(b)*e^(1/2) - A*b^(9/2)*d^3*abs(b)*e^(1/2) - 3*B*a^2*b^(5/2)*d^2*abs(b)*e^(3/2) + 3
*A*a*b^(7/2)*d^2*abs(b)*e^(3/2) + 3*B*a^3*b^(3/2)*d*abs(b)*e^(5/2) - 3*A*a^2*b^(5/2)*d*abs(b)*e^(5/2) - B*a^4*
sqrt(b)*abs(b)*e^(7/2) + A*a^3*b^(3/2)*abs(b)*e^(7/2))/((b^2*d - a*b*e - (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt
(b^2*d + (b*x + a)*b*e - a*b*e))^2)*b^5)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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