3.2230 \(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=167 \[ -\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 167, 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 = {78, 47, 63, 217, 206} \[ -\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 2.79, size = 222, normalized size = 1.33 \[ \frac {2 \left (e^4 (a+b x)^4 (A e-B d)-7 b^2 B e (a+b x) (d+e x)^3 (b d-a e)+\frac {7}{5} B e^3 (a+b x)^3 (d+e x) (a e-b d)-\frac {7}{3} b B e^2 (a+b x)^2 (d+e x)^2 (b d-a e)+\frac {7 B \sqrt {e} \sqrt {a+b x} (b d-a e)^{9/2} \left (\frac {b (d+e x)}{b d-a e}\right )^{7/2} \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{b}\right )}{7 e^5 \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(2*(e^4*(-(B*d) + A*e)*(a + b*x)^4 + (7*B*e^3*(-(b*d) + a*e)*(a + b*x)^3*(d + e*x))/5 - (7*b*B*e^2*(b*d - a*e)
*(a + b*x)^2*(d + e*x)^2)/3 - 7*b^2*B*e*(b*d - a*e)*(a + b*x)*(d + e*x)^3 + (7*B*Sqrt[e]*(b*d - a*e)^(9/2)*Sqr
t[a + b*x]*((b*(d + e*x))/(b*d - a*e))^(7/2)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/b))/(7*e^5*(b*d
- a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2))

________________________________________________________________________________________

fricas [B]  time = 31.38, size = 1053, normalized size = 6.31 \[ \left [\frac {105 \, {\left (B b^{3} d^{5} - B a b^{2} d^{4} e + {\left (B b^{3} d e^{4} - B a b^{2} e^{5}\right )} x^{4} + 4 \, {\left (B b^{3} d^{2} e^{3} - B a b^{2} d e^{4}\right )} x^{3} + 6 \, {\left (B b^{3} d^{3} e^{2} - B a b^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (B b^{3} d^{4} e - B a b^{2} d^{3} e^{2}\right )} x\right )} \sqrt {\frac {b}{e}} \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 e^{2} x + b d e + a e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {b}{e}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (105 \, B b^{3} d^{4} - 70 \, B a b^{2} d^{3} e - 14 \, B a^{2} b d^{2} e^{2} - 6 \, B a^{3} d e^{3} - 15 \, A a^{3} e^{4} + {\left (176 \, B b^{3} d e^{3} - {\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} e^{4}\right )} x^{3} + {\left (406 \, B b^{3} d^{2} e^{2} - 284 \, B a b^{2} d e^{3} - {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} e^{4}\right )} x^{2} + {\left (350 \, B b^{3} d^{3} e - 238 \, B a b^{2} d^{2} e^{2} - 46 \, B a^{2} b d e^{3} - 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{210 \, {\left (b d^{5} e^{4} - a d^{4} e^{5} + {\left (b d e^{8} - a e^{9}\right )} x^{4} + 4 \, {\left (b d^{2} e^{7} - a d e^{8}\right )} x^{3} + 6 \, {\left (b d^{3} e^{6} - a d^{2} e^{7}\right )} x^{2} + 4 \, {\left (b d^{4} e^{5} - a d^{3} e^{6}\right )} x\right )}}, -\frac {105 \, {\left (B b^{3} d^{5} - B a b^{2} d^{4} e + {\left (B b^{3} d e^{4} - B a b^{2} e^{5}\right )} x^{4} + 4 \, {\left (B b^{3} d^{2} e^{3} - B a b^{2} d e^{4}\right )} x^{3} + 6 \, {\left (B b^{3} d^{3} e^{2} - B a b^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (B b^{3} d^{4} e - B a b^{2} d^{3} e^{2}\right )} x\right )} \sqrt {-\frac {b}{e}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {b}{e}}}{2 \, {\left (b^{2} e x^{2} + a b d + {\left (b^{2} d + a b e\right )} x\right )}}\right ) + 2 \, {\left (105 \, B b^{3} d^{4} - 70 \, B a b^{2} d^{3} e - 14 \, B a^{2} b d^{2} e^{2} - 6 \, B a^{3} d e^{3} - 15 \, A a^{3} e^{4} + {\left (176 \, B b^{3} d e^{3} - {\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} e^{4}\right )} x^{3} + {\left (406 \, B b^{3} d^{2} e^{2} - 284 \, B a b^{2} d e^{3} - {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} e^{4}\right )} x^{2} + {\left (350 \, B b^{3} d^{3} e - 238 \, B a b^{2} d^{2} e^{2} - 46 \, B a^{2} b d e^{3} - 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{105 \, {\left (b d^{5} e^{4} - a d^{4} e^{5} + {\left (b d e^{8} - a e^{9}\right )} x^{4} + 4 \, {\left (b d^{2} e^{7} - a d e^{8}\right )} x^{3} + 6 \, {\left (b d^{3} e^{6} - a d^{2} e^{7}\right )} x^{2} + 4 \, {\left (b d^{4} e^{5} - a d^{3} e^{6}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

[1/210*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x
^3 + 6*(B*b^3*d^3*e^2 - B*a*b^2*d^2*e^3)*x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqrt(b/e)*log(8*b^2*e^2*x^
2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e^2*x + b*d*e + a*e^2)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(b/e) + 8*(b
^2*d*e + a*b*e^2)*x) - 4*(105*B*b^3*d^4 - 70*B*a*b^2*d^3*e - 14*B*a^2*b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4
+ (176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e^2 - 284*B*a*b^2*d*e^3 - (77*B*a^2*b
+ 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B*a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)
*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5 + (b*d*e^8 - a*e^9)*x^4 + 4*(b*d^2*e^7 - a*d*e^8)
*x^3 + 6*(b*d^3*e^6 - a*d^2*e^7)*x^2 + 4*(b*d^4*e^5 - a*d^3*e^6)*x), -1/105*(105*(B*b^3*d^5 - B*a*b^2*d^4*e +
(B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^3*d^3*e^2 - B*a*b^2*d^2*e^3)*
x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqrt(-b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x
+ d)*sqrt(-b/e)/(b^2*e*x^2 + a*b*d + (b^2*d + a*b*e)*x)) + 2*(105*B*b^3*d^4 - 70*B*a*b^2*d^3*e - 14*B*a^2*b*d^
2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4 + (176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e
^2 - 284*B*a*b^2*d*e^3 - (77*B*a^2*b + 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B*a*b^2*d^2*e^2 - 46*B*a^
2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5 + (b*d*e^8 -
a*e^9)*x^4 + 4*(b*d^2*e^7 - a*d*e^8)*x^3 + 6*(b*d^3*e^6 - a*d^2*e^7)*x^2 + 4*(b*d^4*e^5 - a*d^3*e^6)*x)]

________________________________________________________________________________________

giac [B]  time = 4.18, size = 635, normalized size = 3.80 \[ -2 \, B b^{\frac {3}{2}} {\left | b \right |} e^{\left (-\frac {9}{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 |}\right ) - \frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left (\frac {{\left (176 \, B b^{10} d^{3} {\left | b \right |} e^{6} - 513 \, B a b^{9} d^{2} {\left | b \right |} e^{7} - 15 \, A b^{10} d^{2} {\left | b \right |} e^{7} + 498 \, B a^{2} b^{8} d {\left | b \right |} e^{8} + 30 \, A a b^{9} d {\left | b \right |} e^{8} - 161 \, B a^{3} b^{7} {\left | b \right |} e^{9} - 15 \, A a^{2} b^{8} {\left | b \right |} e^{9}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}} + \frac {406 \, {\left (B b^{11} d^{4} {\left | b \right |} e^{5} - 4 \, B a b^{10} d^{3} {\left | b \right |} e^{6} + 6 \, B a^{2} b^{9} d^{2} {\left | b \right |} e^{7} - 4 \, B a^{3} b^{8} d {\left | b \right |} e^{8} + B a^{4} b^{7} {\left | b \right |} e^{9}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} + \frac {350 \, {\left (B b^{12} d^{5} {\left | b \right |} e^{4} - 5 \, B a b^{11} d^{4} {\left | b \right |} e^{5} + 10 \, B a^{2} b^{10} d^{3} {\left | b \right |} e^{6} - 10 \, B a^{3} b^{9} d^{2} {\left | b \right |} e^{7} + 5 \, B a^{4} b^{8} d {\left | b \right |} e^{8} - B a^{5} b^{7} {\left | b \right |} e^{9}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} {\left (b x + a\right )} + \frac {105 \, {\left (B b^{13} d^{6} {\left | b \right |} e^{3} - 6 \, B a b^{12} d^{5} {\left | b \right |} e^{4} + 15 \, B a^{2} b^{11} d^{4} {\left | b \right |} e^{5} - 20 \, B a^{3} b^{10} d^{3} {\left | b \right |} e^{6} + 15 \, B a^{4} b^{9} d^{2} {\left | b \right |} e^{7} - 6 \, B a^{5} b^{8} d {\left | b \right |} e^{8} + B a^{6} b^{7} {\left | b \right |} e^{9}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} \sqrt {b x + a}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

-2*B*b^(3/2)*abs(b)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e))) -
2/105*(((b*x + a)*((176*B*b^10*d^3*abs(b)*e^6 - 513*B*a*b^9*d^2*abs(b)*e^7 - 15*A*b^10*d^2*abs(b)*e^7 + 498*B*
a^2*b^8*d*abs(b)*e^8 + 30*A*a*b^9*d*abs(b)*e^8 - 161*B*a^3*b^7*abs(b)*e^9 - 15*A*a^2*b^8*abs(b)*e^9)*(b*x + a)
/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10) + 406*(B*b^11*d^4*abs(b)*e^5 - 4*B*a*b^10*d^
3*abs(b)*e^6 + 6*B*a^2*b^9*d^2*abs(b)*e^7 - 4*B*a^3*b^8*d*abs(b)*e^8 + B*a^4*b^7*abs(b)*e^9)/(b^5*d^3*e^7 - 3*
a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10)) + 350*(B*b^12*d^5*abs(b)*e^4 - 5*B*a*b^11*d^4*abs(b)*e^5 + 10
*B*a^2*b^10*d^3*abs(b)*e^6 - 10*B*a^3*b^9*d^2*abs(b)*e^7 + 5*B*a^4*b^8*d*abs(b)*e^8 - B*a^5*b^7*abs(b)*e^9)/(b
^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10))*(b*x + a) + 105*(B*b^13*d^6*abs(b)*e^3 - 6*B*a
*b^12*d^5*abs(b)*e^4 + 15*B*a^2*b^11*d^4*abs(b)*e^5 - 20*B*a^3*b^10*d^3*abs(b)*e^6 + 15*B*a^4*b^9*d^2*abs(b)*e
^7 - 6*B*a^5*b^8*d*abs(b)*e^8 + B*a^6*b^7*abs(b)*e^9)/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b
^2*e^10))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 1089, normalized size = 6.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x)

[Out]

-1/105*(b*x+a)^(1/2)*(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^4*b^
4*d*e^4+30*A*a^3*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+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))*b^4*d^5-210*B*b^3*d^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+476*B*x*a*b^2*d^2*e^2*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+568*B*x^2*a*b^2*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+92*B*x*a^2*b*d*e^
3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*A*x^2*a*b^2*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+154*B*x^2*a^2*b*e
^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-812*B*x^2*b^3*d^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*A*x*a^2*b*
e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+420*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^3*b^4*d^2*e^3+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^2*b^4*d^3*e^2+420*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*b^4*d^4*e-
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*b^3*d^4*e+30*A*x^3*b^3*e^4
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+42*B*x*a^3*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+12*B*a^3*d*e^3*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)-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^4*a*b^3*e^5-700*B*x*b^3*d^3*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+28*B*a^2*b*d^2*e^2*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)+140*B*a*b^2*d^3*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-420*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^3*a*b^3*d*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^2*a*b^3*d^2*e^3-420*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^3*d^3*e^2+322*B*x^3*a*b^2*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-352*B*x^3*b^3*d*e^3*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/(a*e-b*d)/(b*e)^(1/2)/(e*x+d)^(7/2)/e^4

________________________________________________________________________________________

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)^(5/2)*(B*x+A)/(e*x+d)^(9/2),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?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(9/2),x)

[Out]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(9/2), x)

________________________________________________________________________________________

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)**(5/2)*(B*x+A)/(e*x+d)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________