3.23.54 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx\) [2254]

Optimal. Leaf size=257 \[ \frac {5 e (3 b B d+4 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^4}+\frac {5 e (3 b B d+4 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{6 b^3 (b d-a e)}-\frac {2 (3 b B d+4 A b e-7 a B e) (d+e x)^{5/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {5 \sqrt {e} (b d-a e) (3 b B d+4 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{9/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 49, 52, 65, 223, 212} \begin {gather*} \frac {5 \sqrt {e} (b d-a e) (-7 a B e+4 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{9/2}}+\frac {5 e \sqrt {a+b x} \sqrt {d+e x} (-7 a B e+4 A b e+3 b B d)}{4 b^4}+\frac {5 e \sqrt {a+b x} (d+e x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{6 b^3 (b d-a e)}-\frac {2 (d+e x)^{5/2} (-7 a B e+4 A b e+3 b B d)}{3 b^2 \sqrt {a+b x} (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.14, size = 137, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-b^3 (A b-a B) (d+e x)^3-\frac {(b d-a e)^2 (3 b B d+4 A b e-7 a B e) (a+b x) \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {e (a+b x)}{-b d+a e}\right )}{\sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{3 b^4 (b d-a e) (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-(b^3*(A*b - a*B)*(d + e*x)^3) - ((b*d - a*e)^2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(a + b*x)*Hype
rgeometric2F1[-5/2, -1/2, 1/2, (e*(a + b*x))/(-(b*d) + a*e)])/Sqrt[(b*(d + e*x))/(b*d - a*e)]))/(3*b^4*(b*d -
a*e)*(a + b*x)^(3/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1249\) vs. \(2(219)=438\).
time = 0.10, size = 1250, normalized size = 4.86

method	result	size
default	\(\text {Expression too large to display}\)	\(1250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(e*x+d)^(1/2)*(-210*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-60*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+150*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^2-45*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-12*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)+48*B*b^3*d^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)-120*A*a^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+32*B*a*b^2*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+
120*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+60*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*e^3*x^2-60*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*e^2*x^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^2*b^2*e^3*x^2-45*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^2*e*x^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^4*e^3+150*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*e
^2*x^2-316*B*a*b^2*d*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-120*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+300*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+16*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)+60*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^3-90*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+42*B*a*b^2
*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-54*B*b^3*d*e*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-160*A*a*b^2*
e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+112*A*b^3*d*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+280*B*a^2*b*e^2*
x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+80*A*a*b^2*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-230*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)^(3/2)/b^4

________________________________________________________________________________________

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)^(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(b*d-%e*a>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [A]
time = 2.47, size = 864, normalized size = 3.36 \begin {gather*} \left [\frac {\frac {15 \, {\left (3 \, B b^{4} d^{2} x^{2} + 6 \, B a b^{3} d^{2} x + 3 \, B a^{2} b^{2} d^{2} + {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} e^{2} - 2 \, {\left ({\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d x^{2} + 2 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d x + {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d\right )} e\right )} e^{\frac {1}{2}} \log \left (b^{2} d^{2} + \frac {4 \, {\left (b^{2} d + {\left (2 \, b^{2} x + a b\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} e^{\frac {1}{2}}}{\sqrt {b}} + {\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 )}{\sqrt {b}} - 4 \, {\left (24 \, B b^{3} d^{2} x + 8 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d^{2} - {\left (6 \, B b^{3} x^{3} - 105 \, B a^{3} + 60 \, A a^{2} b - 3 \, {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{2} - 20 \, {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} e^{2} - {\left (27 \, B b^{3} d x^{2} + 2 \, {\left (79 \, B a b^{2} - 28 \, A b^{3}\right )} d x + 5 \, {\left (23 \, B a^{2} b - 8 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{48 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (3 \, B b^{4} d^{2} x^{2} + 6 \, B a b^{3} d^{2} x + 3 \, B a^{2} b^{2} d^{2} + {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} e^{2} - 2 \, {\left ({\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d x^{2} + 2 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d x + {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d\right )} e\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {-\frac {e}{b}}}{2 \, {\left ({\left (b x^{2} + a x\right )} e^{2} + {\left (b d x + a d\right )} e\right )}}\right ) + 2 \, {\left (24 \, B b^{3} d^{2} x + 8 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d^{2} - {\left (6 \, B b^{3} x^{3} - 105 \, B a^{3} + 60 \, A a^{2} b - 3 \, {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{2} - 20 \, {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} e^{2} - {\left (27 \, B b^{3} d x^{2} + 2 \, {\left (79 \, B a b^{2} - 28 \, A b^{3}\right )} d x + 5 \, {\left (23 \, B a^{2} b - 8 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\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)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*(3*B*b^4*d^2*x^2 + 6*B*a*b^3*d^2*x + 3*B*a^2*b^2*d^2 + (7*B*a^4 - 4*A*a^3*b + (7*B*a^2*b^2 - 4*A*a*b
^3)*x^2 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x)*e^2 - 2*((5*B*a*b^3 - 2*A*b^4)*d*x^2 + 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*
x + (5*B*a^3*b - 2*A*a^2*b^2)*d)*e)*e^(1/2)*log(b^2*d^2 + 4*(b^2*d + (2*b^2*x + a*b)*e)*sqrt(b*x + a)*sqrt(x*e
+ d)*e^(1/2)/sqrt(b) + (8*b^2*x^2 + 8*a*b*x + a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e)/sqrt(b) - 4*(24*B*b^3*d^2
*x + 8*(2*B*a*b^2 + A*b^3)*d^2 - (6*B*b^3*x^3 - 105*B*a^3 + 60*A*a^2*b - 3*(7*B*a*b^2 - 4*A*b^3)*x^2 - 20*(7*B
*a^2*b - 4*A*a*b^2)*x)*e^2 - (27*B*b^3*d*x^2 + 2*(79*B*a*b^2 - 28*A*b^3)*d*x + 5*(23*B*a^2*b - 8*A*a*b^2)*d)*e
)*sqrt(b*x + a)*sqrt(x*e + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/24*(15*(3*B*b^4*d^2*x^2 + 6*B*a*b^3*d^2*x +
3*B*a^2*b^2*d^2 + (7*B*a^4 - 4*A*a^3*b + (7*B*a^2*b^2 - 4*A*a*b^3)*x^2 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x)*e^2 -
2*((5*B*a*b^3 - 2*A*b^4)*d*x^2 + 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*x + (5*B*a^3*b - 2*A*a^2*b^2)*d)*e)*sqrt(-e/b)
*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(-e/b)/((b*x^2 + a*x)*e^2 + (b*d*x + a*d)*e)
) + 2*(24*B*b^3*d^2*x + 8*(2*B*a*b^2 + A*b^3)*d^2 - (6*B*b^3*x^3 - 105*B*a^3 + 60*A*a^2*b - 3*(7*B*a*b^2 - 4*A
*b^3)*x^2 - 20*(7*B*a^2*b - 4*A*a*b^2)*x)*e^2 - (27*B*b^3*d*x^2 + 2*(79*B*a*b^2 - 28*A*b^3)*d*x + 5*(23*B*a^2*
b - 8*A*a*b^2)*d)*e)*sqrt(b*x + a)*sqrt(x*e + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1287 vs. \(2 (236) = 472\).
time = 0.91, size = 1287, normalized size = 5.01 \begin {gather*} \frac {1}{4} \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |} e^{2}}{b^{6}} + \frac {{\left (9 \, B b^{12} d {\left | b \right |} e^{3} - 13 \, B a b^{11} {\left | b \right |} e^{4} + 4 \, A b^{12} {\left | b \right |} e^{4}\right )} e^{\left (-2\right )}}{b^{17}}\right )} - \frac {5 \, {\left (3 \, B b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {1}{2}} - 10 \, B a b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 4 \, A b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 7 \, B a^{2} \sqrt {b} {\left | b \right |} e^{\frac {5}{2}} - 4 \, A a b^{\frac {3}{2}} {\left | b \right |} e^{\frac {5}{2}}\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 )}{8 \, b^{6}} - \frac {4 \, {\left (3 \, B b^{\frac {15}{2}} d^{5} {\left | b \right |} e^{\frac {1}{2}} - 22 \, B a b^{\frac {13}{2}} d^{4} {\left | b \right |} e^{\frac {3}{2}} + 7 \, A b^{\frac {15}{2}} d^{4} {\left | b \right |} e^{\frac {3}{2}} - 6 \, {\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} B b^{\frac {11}{2}} d^{4} {\left | b \right |} e^{\frac {1}{2}} + 58 \, B a^{2} b^{\frac {11}{2}} d^{3} {\left | b \right |} e^{\frac {5}{2}} - 28 \, A a b^{\frac {13}{2}} d^{3} {\left | b \right |} e^{\frac {5}{2}} + 36 \, {\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} B a b^{\frac {9}{2}} d^{3} {\left | b \right |} e^{\frac {3}{2}} - 12 \, {\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} A b^{\frac {11}{2}} d^{3} {\left | b \right |} e^{\frac {3}{2}} + 3 \, {\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 )}^{4} B b^{\frac {7}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - 72 \, B a^{3} b^{\frac {9}{2}} d^{2} {\left | b \right |} e^{\frac {7}{2}} + 42 \, A a^{2} b^{\frac {11}{2}} d^{2} {\left | b \right |} e^{\frac {7}{2}} - 72 \, {\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} B a^{2} b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {5}{2}} + 36 \, {\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} A a b^{\frac {9}{2}} d^{2} {\left | b \right |} e^{\frac {5}{2}} - 18 \, {\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 )}^{4} B a b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 9 \, {\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 )}^{4} A b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 43 \, B a^{4} b^{\frac {7}{2}} d {\left | b \right |} e^{\frac {9}{2}} - 28 \, A a^{3} b^{\frac {9}{2}} d {\left | b \right |} e^{\frac {9}{2}} + 60 \, {\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} B a^{3} b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {7}{2}} - 36 \, {\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} A a^{2} b^{\frac {7}{2}} d {\left | b \right |} e^{\frac {7}{2}} + 27 \, {\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 )}^{4} B a^{2} b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 18 \, {\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 )}^{4} A a b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 10 \, B a^{5} b^{\frac {5}{2}} {\left | b \right |} e^{\frac {11}{2}} + 7 \, A a^{4} b^{\frac {7}{2}} {\left | b \right |} e^{\frac {11}{2}} - 18 \, {\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} B a^{4} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {9}{2}} + 12 \, {\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} A a^{3} b^{\frac {5}{2}} {\left | b \right |} e^{\frac {9}{2}} - 12 \, {\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 )}^{4} B a^{3} \sqrt {b} {\left | b \right |} e^{\frac {7}{2}} + 9 \, {\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 )}^{4} A a^{2} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {7}{2}}\right )}}{3 \, {\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 )}^{3} 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)^(5/2),x, algorithm="giac")

[Out]

1/4*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*B*abs(b)*e^2/b^6 + (9*B*b^12*d*abs(b)*e^3 -
13*B*a*b^11*abs(b)*e^4 + 4*A*b^12*abs(b)*e^4)*e^(-2)/b^17) - 5/8*(3*B*b^(5/2)*d^2*abs(b)*e^(1/2) - 10*B*a*b^(
3/2)*d*abs(b)*e^(3/2) + 4*A*b^(5/2)*d*abs(b)*e^(3/2) + 7*B*a^2*sqrt(b)*abs(b)*e^(5/2) - 4*A*a*b^(3/2)*abs(b)*e
^(5/2))*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 - 4/3*(3*B*b^(15/2)*d
^5*abs(b)*e^(1/2) - 22*B*a*b^(13/2)*d^4*abs(b)*e^(3/2) + 7*A*b^(15/2)*d^4*abs(b)*e^(3/2) - 6*(sqrt(b*x + a)*sq
rt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^(11/2)*d^4*abs(b)*e^(1/2) + 58*B*a^2*b^(11/2)*d^3*a
bs(b)*e^(5/2) - 28*A*a*b^(13/2)*d^3*abs(b)*e^(5/2) + 36*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a
)*b*e - a*b*e))^2*B*a*b^(9/2)*d^3*abs(b)*e^(3/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*
b*e - a*b*e))^2*A*b^(11/2)*d^3*abs(b)*e^(3/2) + 3*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e
- a*b*e))^4*B*b^(7/2)*d^3*abs(b)*e^(1/2) - 72*B*a^3*b^(9/2)*d^2*abs(b)*e^(7/2) + 42*A*a^2*b^(11/2)*d^2*abs(b)*
e^(7/2) - 72*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^2*b^(7/2)*d^2*abs(b)*
e^(5/2) + 36*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a*b^(9/2)*d^2*abs(b)*e^
(5/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a*b^(5/2)*d^2*abs(b)*e^(3
/2) + 9*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^(7/2)*d^2*abs(b)*e^(3/2) +
43*B*a^4*b^(7/2)*d*abs(b)*e^(9/2) - 28*A*a^3*b^(9/2)*d*abs(b)*e^(9/2) + 60*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - s
qrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^3*b^(5/2)*d*abs(b)*e^(7/2) - 36*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqr
t(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a^2*b^(7/2)*d*abs(b)*e^(7/2) + 27*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(
b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^2*b^(3/2)*d*abs(b)*e^(5/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^
2*d + (b*x + a)*b*e - a*b*e))^4*A*a*b^(5/2)*d*abs(b)*e^(5/2) - 10*B*a^5*b^(5/2)*abs(b)*e^(11/2) + 7*A*a^4*b^(7
/2)*abs(b)*e^(11/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^4*b^(3/2)
*abs(b)*e^(9/2) + 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a^3*b^(5/2)*abs
(b)*e^(9/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^3*sqrt(b)*abs(b)*
e^(7/2) + 9*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*a^2*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)^3*b^5)

________________________________________________________________________________________

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 )}^{5/2}} \,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)^(5/2),x)

[Out]

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

________________________________________________________________________________________