3.22.15 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=257 \[ \frac {5 \sqrt {e} (b d-a e) (-7 a B e+4 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{9/2}}+\frac {5 e \sqrt {a+b x} \sqrt {d+e x} (-7 a B e+4 A b e+3 b B d)}{4 b^4}+\frac {5 e \sqrt {a+b x} (d+e x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{6 b^3 (b d-a e)}-\frac {2 (d+e x)^{5/2} (-7 a B e+4 A b e+3 b B d)}{3 b^2 \sqrt {a+b x} (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.20, 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 =
{78, 47, 50, 63, 217, 206} \begin {gather*} -\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 {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 {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 \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 {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 47
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 50
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 63
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 78
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] || LtQ
[p, n]))))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 217
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)) \operatorname {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)) \operatorname {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*}
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Mathematica [C] time = 0.16, size = 137, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-b^3 (d+e x)^3 (A b-a B)-\frac {(a+b x) (b d-a e)^2 (-7 a B e+4 A b e+3 b B d) \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {e (a+b x)}{a e-b d}\right )}{\sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{3 b^4 (a+b x)^{3/2} (b d-a e)} \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))
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IntegrateAlgebraic [F] time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(5/2),x]
[Out]
$Aborted
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fricas [A] time = 7.65, size = 887, normalized size = 3.45 \begin {gather*} \left [\frac {15 \, {\left (3 \, B a^{2} b^{2} d^{2} - 2 \, {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e + {\left (7 \, B a^{4} - 4 \, A a^{3} b\right )} e^{2} + {\left (3 \, B b^{4} d^{2} - 2 \, {\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d e + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (3 \, B a b^{3} d^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e + {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} e^{2}\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 (6 \, B b^{3} e^{2} x^{3} - 8 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (23 \, B a^{2} b - 8 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + 3 \, {\left (9 \, B b^{3} d e - {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{2} - 2 \, {\left (12 \, B b^{3} d^{2} - {\left (79 \, B a b^{2} - 28 \, A b^{3}\right )} d e + 10 \, {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (3 \, B a^{2} b^{2} d^{2} - 2 \, {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e + {\left (7 \, B a^{4} - 4 \, A a^{3} b\right )} e^{2} + {\left (3 \, B b^{4} d^{2} - 2 \, {\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d e + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (3 \, B a b^{3} d^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e + {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} e^{2}\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 (6 \, B b^{3} e^{2} x^{3} - 8 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (23 \, B a^{2} b - 8 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + 3 \, {\left (9 \, B b^{3} d e - {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{2} - 2 \, {\left (12 \, B b^{3} d^{2} - {\left (79 \, B a b^{2} - 28 \, A b^{3}\right )} d e + 10 \, {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + 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*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^3*b)*e^2 + (3*B*b^4*d^2 - 2*(5
*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*A*a*b^3)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*
d*e + (7*B*a^3*b - 4*A*a^2*b^2)*e^2)*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*(6*B*b^3*e^2*x^3 -
8*(2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4*A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (
7*B*a*b^2 - 4*A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*e + 10*(7*B*a^2*b - 4*A*a*b^2)*e^2
)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/24*(15*(3*B*a^2*b^2*d^2 - 2*(5*B*a^3*b -
2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^3*b)*e^2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*
A*a*b^3)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*e + (7*B*a^3*b - 4*A*a^2*b^2)*e^2)*x)*sqr
t(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(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*(6*B*b^3*e^2*x^3 - 8*(2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4
*A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (7*B*a*b^2 - 4*A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*
e + 10*(7*B*a^2*b - 4*A*a*b^2)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]
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giac [B] time = 3.92, size = 1287, normalized size = 5.01
result too large to display
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)
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maple [B] time = 0.02, size = 1250, normalized size = 4.86
result too large to display
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)
[Out]
-1/24*(e*x+d)^(1/2)*(-12*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^3*e^2*x^3+60*A*a^3*b*e^3*ln(1/2*(2*b*e*x+a*e+
b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-105*B*a^4*e^3*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+
d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-316*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a*b^2*d*e*x+16*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)*A*b^3*d^2+210*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a^3*e^2+48*((b*x+a)*(e*x+d))^(1/2)*(b*e
)^(1/2)*B*b^3*d^2*x-120*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*a^2*b*e^2+32*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
*B*a*b^2*d^2+120*A*a^2*b^2*e^3*x*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-2
10*B*a^3*b*e^3*x*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-60*A*a^2*b^2*d*e^
2*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+150*B*a^3*b*d*e^2*ln(1/2*(2*b*e*
x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-45*B*a^2*b^2*d^2*e*ln(1/2*(2*b*e*x+a*e+b*d+2*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-24*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*b^3*e^2*x^2+150*B*ln(1
/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*x^2*a*b^3*d*e^2-230*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)*B*a^2*b*d*e-120*A*a*b^3*d*e^2*x*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2
))/(b*e)^(1/2))+300*B*a^2*b^2*d*e^2*x*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/
2))-90*B*a*b^3*d^2*e*x*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+42*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a*b^2*e^2*x^2-54*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^3*d*e*x^2-160*((b*x+a)*
(e*x+d))^(1/2)*(b*e)^(1/2)*A*a*b^2*e^2*x+112*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*b^3*d*e*x+280*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)*B*a^2*b*e^2*x+80*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*a*b^2*d*e+60*A*ln(1/2*(2*b*e*x+a
*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*x^2*a*b^3*e^3-60*A*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+
a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*x^2*b^4*d*e^2-105*B*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1
/2)*(b*e)^(1/2))/(b*e)^(1/2))*x^2*a^2*b^2*e^3-45*B*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/
2))/(b*e)^(1/2))*x^2*b^4*d^2*e)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(3/2)/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)^(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 details)Is a*e-b*d zero or nonzero?
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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 )}^{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)
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sympy [F(-1)] 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
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