3.2253 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=302 \[ \frac {35 \sqrt {e} (b d-a e)^2 (-3 a B e+2 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}}+\frac {35 e \sqrt {a+b x} \sqrt {d+e x} (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^5}+\frac {35 e \sqrt {a+b x} (d+e x)^{3/2} (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac {7 e \sqrt {a+b x} (d+e x)^{5/2} (-3 a B e+2 A b e+b B d)}{3 b^3 (b d-a e)}-\frac {2 (d+e x)^{7/2} (-3 a B e+2 A b e+b B d)}{b^2 \sqrt {a+b x} (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

[Out]

-2/3*(A*b-B*a)*(e*x+d)^(9/2)/b/(-a*e+b*d)/(b*x+a)^(3/2)+35/8*(-a*e+b*d)^2*(2*A*b*e-3*B*a*e+B*b*d)*arctanh(e^(1
/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))*e^(1/2)/b^(11/2)-2*(2*A*b*e-3*B*a*e+B*b*d)*(e*x+d)^(7/2)/b^2/(-a*e+b*
d)/(b*x+a)^(1/2)+35/12*e*(2*A*b*e-3*B*a*e+B*b*d)*(e*x+d)^(3/2)*(b*x+a)^(1/2)/b^4+7/3*e*(2*A*b*e-3*B*a*e+B*b*d)
*(e*x+d)^(5/2)*(b*x+a)^(1/2)/b^3/(-a*e+b*d)+35/8*e*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1
/2)/b^5

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Rubi [A]  time = 0.24, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {2 (d+e x)^{7/2} (-3 a B e+2 A b e+b B d)}{b^2 \sqrt {a+b x} (b d-a e)}+\frac {7 e \sqrt {a+b x} (d+e x)^{5/2} (-3 a B e+2 A b e+b B d)}{3 b^3 (b d-a e)}+\frac {35 e \sqrt {a+b x} (d+e x)^{3/2} (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac {35 e \sqrt {a+b x} \sqrt {d+e x} (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^5}+\frac {35 \sqrt {e} (b d-a e)^2 (-3 a B e+2 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x]

[Out]

(35*e*(b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^5) + (35*e*(b*B*d + 2*A*b*e -
3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^4) + (7*e*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5
/2))/(3*b^3*(b*d - a*e)) - (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(b^2*(b*d - a*e)*Sqrt[a + b*x]) - (
2*(A*b - a*B)*(d + e*x)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (35*Sqrt[e]*(b*d - a*e)^2*(b*B*d + 2*A*b*e
- 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(11/2))

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)^{7/2}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(b B d+2 A b e-3 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^{3/2}} \, dx}{b (b d-a e)}\\ &=-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(7 e (b B d+2 A b e-3 a B e)) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}} \, dx}{b^2 (b d-a e)}\\ &=\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(35 e (b B d+2 A b e-3 a B e)) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^3}\\ &=\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(35 e (b d-a e) (b B d+2 A b e-3 a B e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{8 b^4}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\left (35 e (b d-a e)^2 (b B d+2 A b e-3 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{16 b^5}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\left (35 e (b d-a e)^2 (b B d+2 A b e-3 a B e)\right ) \operatorname {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^6}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\left (35 e (b d-a e)^2 (b B d+2 A b e-3 a B e)\right ) \operatorname {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^6}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}}\\ \end {align*}

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Mathematica [C]  time = 0.14, size = 136, normalized size = 0.45 \[ \frac {2 \sqrt {d+e x} \left (-b^4 (d+e x)^4 (A b-a B)-\frac {3 (a+b x) (b d-a e)^3 (-3 a B e+2 A b e+b B d) \, _2F_1\left (-\frac {7}{2},-\frac {1}{2};\frac {1}{2};\frac {e (a+b x)}{a e-b d}\right )}{\sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{3 b^5 (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x]

[Out]

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

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fricas [B]  time = 5.80, size = 1271, normalized size = 4.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/96*(105*(B*a^2*b^3*d^3 - (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4*b - 4*A*a^3*b^2)*d*e^2 - (3*B*a^5 -
2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b^4 - 2*A*b^5)*d^2*e + (7*B*a^2*b^3 - 4*A*a*b^4)*d*e^2 - (3*B*a^3*b^2 - 2
*A*a^2*b^3)*e^3)*x^2 + 2*(B*a*b^4*d^3 - (5*B*a^2*b^3 - 2*A*a*b^4)*d^2*e + (7*B*a^3*b^2 - 4*A*a^2*b^3)*d*e^2 -
(3*B*a^4*b - 2*A*a^3*b^2)*e^3)*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*(8*B*b^4*e^3*x^4 - 16*(2*
B*a*b^3 + A*b^4)*d^3 + 7*(49*B*a^2*b^2 - 16*A*a*b^3)*d^2*e - 70*(9*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(3*B*a^4
- 2*A*a^3*b)*e^3 + 2*(19*B*b^4*d*e^2 - 3*(3*B*a*b^3 - 2*A*b^4)*e^3)*x^3 + 3*(29*B*b^4*d^2*e - 2*(23*B*a*b^3 -
13*A*b^4)*d*e^2 + 7*(3*B*a^2*b^2 - 2*A*a*b^3)*e^3)*x^2 - 2*(24*B*b^4*d^3 - (239*B*a*b^3 - 80*A*b^4)*d^2*e + 7
*(61*B*a^2*b^2 - 34*A*a*b^3)*d*e^2 - 70*(3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^7*x^
2 + 2*a*b^6*x + a^2*b^5), -1/48*(105*(B*a^2*b^3*d^3 - (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4*b - 4*A*a^3
*b^2)*d*e^2 - (3*B*a^5 - 2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b^4 - 2*A*b^5)*d^2*e + (7*B*a^2*b^3 - 4*A*a*b^4)
*d*e^2 - (3*B*a^3*b^2 - 2*A*a^2*b^3)*e^3)*x^2 + 2*(B*a*b^4*d^3 - (5*B*a^2*b^3 - 2*A*a*b^4)*d^2*e + (7*B*a^3*b^
2 - 4*A*a^2*b^3)*d*e^2 - (3*B*a^4*b - 2*A*a^3*b^2)*e^3)*x)*sqrt(-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*(8*B*b^4*e^3*x^4 - 16*(2*B*a*b^3
+ A*b^4)*d^3 + 7*(49*B*a^2*b^2 - 16*A*a*b^3)*d^2*e - 70*(9*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(3*B*a^4 - 2*A*a
^3*b)*e^3 + 2*(19*B*b^4*d*e^2 - 3*(3*B*a*b^3 - 2*A*b^4)*e^3)*x^3 + 3*(29*B*b^4*d^2*e - 2*(23*B*a*b^3 - 13*A*b^
4)*d*e^2 + 7*(3*B*a^2*b^2 - 2*A*a*b^3)*e^3)*x^2 - 2*(24*B*b^4*d^3 - (239*B*a*b^3 - 80*A*b^4)*d^2*e + 7*(61*B*a
^2*b^2 - 34*A*a*b^3)*d*e^2 - 70*(3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^7*x^2 + 2*a*
b^6*x + a^2*b^5)]

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giac [B]  time = 5.51, size = 1654, normalized size = 5.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),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^3/b^7 + (19*B*b^21
*d*abs(b)*e^6 - 25*B*a*b^20*abs(b)*e^7 + 6*A*b^21*abs(b)*e^7)*e^(-4)/b^27) + 3*(29*B*b^22*d^2*abs(b)*e^5 - 84*
B*a*b^21*d*abs(b)*e^6 + 26*A*b^22*d*abs(b)*e^6 + 55*B*a^2*b^20*abs(b)*e^7 - 26*A*a*b^21*abs(b)*e^7)*e^(-4)/b^2
7) - 35/16*(B*b^(7/2)*d^3*abs(b)*e^(1/2) - 5*B*a*b^(5/2)*d^2*abs(b)*e^(3/2) + 2*A*b^(7/2)*d^2*abs(b)*e^(3/2) +
7*B*a^2*b^(3/2)*d*abs(b)*e^(5/2) - 4*A*a*b^(5/2)*d*abs(b)*e^(5/2) - 3*B*a^3*sqrt(b)*abs(b)*e^(7/2) + 2*A*a^2*
b^(3/2)*abs(b)*e^(7/2))*log((sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/b^7 - 4/3
*(3*B*b^(17/2)*d^6*abs(b)*e^(1/2) - 28*B*a*b^(15/2)*d^5*abs(b)*e^(3/2) + 10*A*b^(17/2)*d^5*abs(b)*e^(3/2) - 6*
(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^(13/2)*d^5*abs(b)*e^(1/2) + 95*B*a
^2*b^(13/2)*d^4*abs(b)*e^(5/2) - 50*A*a*b^(15/2)*d^4*abs(b)*e^(5/2) + 48*(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^(11/2)*d^4*abs(b)*e^(3/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(
b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^(13/2)*d^4*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^(9/2)*d^4*abs(b)*e^(1/2) - 160*B*a^3*b^(11/2)*d^3*abs(b)*e^(7/2) + 100*A*a^2
*b^(13/2)*d^3*abs(b)*e^(7/2) - 132*(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^(9/2)*d^3*abs(b)*e^(5/2) + 72*(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^(11/2)*d^3*abs(b)*e^(5/2) - 24*(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^(7/2)*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))^4*A*b^(9
/2)*d^3*abs(b)*e^(3/2) + 145*B*a^4*b^(9/2)*d^2*abs(b)*e^(9/2) - 100*A*a^3*b^(11/2)*d^2*abs(b)*e^(9/2) + 168*(s
qrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^3*b^(7/2)*d^2*abs(b)*e^(7/2) - 108*(
sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a^2*b^(9/2)*d^2*abs(b)*e^(7/2) + 54*(
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^(5/2)*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))^4*A*a*b^(7/2)*d^2*abs(b)*e^(5/2) - 68*B*a
^5*b^(7/2)*d*abs(b)*e^(11/2) + 50*A*a^4*b^(9/2)*d*abs(b)*e^(11/2) - 102*(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^(5/2)*d*abs(b)*e^(9/2) + 72*(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^(7/2)*d*abs(b)*e^(9/2) - 48*(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*b^(3/2)*d*abs(b)*e^(7/2) + 36*(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^(5/2)*d*abs(b)*e^(7/2) + 13*B*a^6*b^(5/2)*abs(b)*e^(13/2) - 10*A*a^5*b^(7/
2)*abs(b)*e^(13/2) + 24*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^5*b^(3/2)*
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*A*a^4*b^(5/2)*abs
(b)*e^(11/2) + 15*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^4*sqrt(b)*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*A*a^3*b^(3/2)*abs(b)*e^(
9/2))/((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^6)

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maple [B]  time = 0.04, size = 1882, normalized size = 6.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),x)

[Out]

1/48*(e*x+d)^(1/2)*(-315*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))*a^5*e^4
-32*A*b^4*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-276*B*x^2*a*b^3*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+21
0*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))*a^4*b*e^4+735*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))*a^4*b*d*e^3+630*B*a^4*e^3*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)-420*A*a^3*b*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-64*B*a*b^3*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)+210*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^2*b^3*e^4+210*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^5*d^2*e^2-315*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^3*b^2*e^4+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*b^5*d^3*e+420*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*a^3*b^2*e^4-630*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*a^4*b*e^4-420*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))*a^3*b^2*d*e^3+210*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))*a^2*b
^3*d^2*e^2+952*A*x*a*b^3*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-1708*B*x*a^2*b^2*d*e^2*((b*x+a)*(e*x+d))^(1
/2)*(b*e)^(1/2)+956*B*x*a*b^3*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-525*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))*a^3*b^2*d^2*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))*a^2*b^3*d^3*e+16*B*x^4*b^4*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+24*A*x^3*b^
4*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*B*x*b^4*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-36*B*x^3*a*b^3*e^
3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+76*B*x^3*b^4*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-84*A*x^2*a*b^3*e^
3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+156*A*x^2*b^4*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+126*B*x^2*a^2*b^
2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+174*B*x^2*b^4*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-560*A*x*a^2*
b^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-320*A*x*b^4*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+840*B*x*a^3*
b*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+700*A*a^2*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-224*A*a*b^3*
d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-1260*B*a^3*b*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+686*B*a^2*b^2
*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-525*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^4*d^2*e^2-840*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*a^2*b^3*d*e^3+420*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*a*b^4
*d^2*e^2+1470*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*a^3*b^2*d*e^3-10
50*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*a^2*b^3*d^2*e^2+210*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*a*b^4*d^3*e-420*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^4*d*e^3+735*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^3*d*e^3)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(3
/2)/b^5

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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