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

Optimal. Leaf size=256 \[ -\frac {b^{3/2} (-5 a B e-2 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}+\frac {b^2 \sqrt {a+b x} \sqrt {d+e x} (-5 a B e-2 A b e+7 b B d)}{e^4 (b d-a e)}-\frac {2 b (a+b x)^{3/2} (-5 a B e-2 A b e+7 b B d)}{3 e^3 \sqrt {d+e x} (b d-a e)}-\frac {2 (a+b x)^{5/2} (-5 a B e-2 A b e+7 b B d)}{15 e^2 (d+e x)^{3/2} (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]

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Rubi [A]  time = 0.20, antiderivative size = 256, 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 {b^2 \sqrt {a+b x} \sqrt {d+e x} (-5 a B e-2 A b e+7 b B d)}{e^4 (b d-a e)}-\frac {b^{3/2} (-5 a B e-2 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}-\frac {2 (a+b x)^{5/2} (-5 a B e-2 A b e+7 b B d)}{15 e^2 (d+e x)^{3/2} (b d-a e)}-\frac {2 b (a+b x)^{3/2} (-5 a B e-2 A b e+7 b B d)}{3 e^3 \sqrt {d+e x} (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) - (2*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(a + b*x
)^(5/2))/(15*e^2*(b*d - a*e)*(d + e*x)^(3/2)) - (2*b*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(a + b*x)^(3/2))/(3*e^3*(b*
d - a*e)*Sqrt[d + e*x]) + (b^2*(7*b*B*d - 2*A*b*e - 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(e^4*(b*d - a*e)) -
(b^(3/2)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*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 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)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {(7 b B d-2 A b e-5 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{5/2}} \, dx}{5 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}+\frac {(b (7 b B d-2 A b e-5 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e^2 (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {\left (b^2 (7 b B d-2 A b e-5 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{e^3 (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {\left (b^2 (7 b B d-2 A b e-5 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{2 e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {(b (7 b B d-2 A b e-5 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 )}{e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {(b (7 b B d-2 A b e-5 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 )}{e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {b^{3/2} (7 b B d-2 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.24, size = 113, normalized size = 0.44 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (-\frac {\left (\frac {b (d+e x)}{b d-a e}\right )^{5/2} (-5 a B e-2 A b e+7 b B d) \, _2F_1\left (\frac {5}{2},\frac {7}{2};\frac {9}{2};\frac {e (a+b x)}{a e-b d}\right )}{b}-7 A e+7 B d\right )}{35 e (d+e x)^{5/2} (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(7/2)*(7*B*d - 7*A*e - ((7*b*B*d - 2*A*b*e - 5*a*B*e)*((b*(d + e*x))/(b*d - a*e))^(5/2)*Hypergeom
etric2F1[5/2, 7/2, 9/2, (e*(a + b*x))/(-(b*d) + a*e)])/b))/(35*e*(-(b*d) + a*e)*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.43, size = 313, normalized size = 1.22 \begin {gather*} \frac {\left (5 a b^{3/2} B e+2 A b^{5/2} e-7 b^{5/2} B d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{9/2}}+\frac {(a+b x)^{5/2} \left (\frac {30 A b^3 e (d+e x)^3}{(a+b x)^3}-\frac {20 A b^2 e^2 (d+e x)^2}{(a+b x)^2}-\frac {4 A b e^3 (d+e x)}{a+b x}-\frac {105 b^3 B d (d+e x)^3}{(a+b x)^3}+\frac {75 a b^2 B e (d+e x)^3}{(a+b x)^3}+\frac {70 b^2 B d e (d+e x)^2}{(a+b x)^2}-\frac {10 a B e^3 (d+e x)}{a+b x}-\frac {50 a b B e^2 (d+e x)^2}{(a+b x)^2}+\frac {14 b B d e^2 (d+e x)}{a+b x}-6 A e^4+6 B d e^3\right )}{15 e^4 (d+e x)^{5/2} \left (e-\frac {b (d+e x)}{a+b x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(5/2)*(6*B*d*e^3 - 6*A*e^4 + (14*b*B*d*e^2*(d + e*x))/(a + b*x) - (4*A*b*e^3*(d + e*x))/(a + b*x) -
(10*a*B*e^3*(d + e*x))/(a + b*x) + (70*b^2*B*d*e*(d + e*x)^2)/(a + b*x)^2 - (20*A*b^2*e^2*(d + e*x)^2)/(a + b
*x)^2 - (50*a*b*B*e^2*(d + e*x)^2)/(a + b*x)^2 - (105*b^3*B*d*(d + e*x)^3)/(a + b*x)^3 + (30*A*b^3*e*(d + e*x)
^3)/(a + b*x)^3 + (75*a*b^2*B*e*(d + e*x)^3)/(a + b*x)^3))/(15*e^4*(d + e*x)^(5/2)*(e - (b*(d + e*x))/(a + b*x
))) + ((-7*b^(5/2)*B*d + 2*A*b^(5/2)*e + 5*a*b^(3/2)*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*
x])])/e^(9/2)

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fricas [A]  time = 19.06, size = 835, normalized size = 3.26 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{2} d^{4} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (7 \, B b^{2} d e^{3} - {\left (5 \, B a b + 2 \, A b^{2}\right )} e^{4}\right )} x^{3} + 3 \, {\left (7 \, B b^{2} d^{2} e^{2} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3}\right )} x^{2} + 3 \, {\left (7 \, B b^{2} d^{3} e - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} 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 (15 \, B b^{2} e^{3} x^{3} + 105 \, B b^{2} d^{3} - 6 \, A a^{2} e^{3} - 10 \, {\left (4 \, B a b + 3 \, A b^{2}\right )} d^{2} e - 2 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} + {\left (161 \, B b^{2} d e^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} e^{3}\right )} x^{2} + {\left (245 \, B b^{2} d^{2} e - 14 \, {\left (7 \, B a b + 5 \, A b^{2}\right )} d e^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{60 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}}, \frac {15 \, {\left (7 \, B b^{2} d^{4} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (7 \, B b^{2} d e^{3} - {\left (5 \, B a b + 2 \, A b^{2}\right )} e^{4}\right )} x^{3} + 3 \, {\left (7 \, B b^{2} d^{2} e^{2} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3}\right )} x^{2} + 3 \, {\left (7 \, B b^{2} d^{3} e - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} 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 (15 \, B b^{2} e^{3} x^{3} + 105 \, B b^{2} d^{3} - 6 \, A a^{2} e^{3} - 10 \, {\left (4 \, B a b + 3 \, A b^{2}\right )} d^{2} e - 2 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} + {\left (161 \, B b^{2} d e^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} e^{3}\right )} x^{2} + {\left (245 \, B b^{2} d^{2} e - 14 \, {\left (7 \, B a b + 5 \, A b^{2}\right )} d e^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{30 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/60*(15*(7*B*b^2*d^4 - (5*B*a*b + 2*A*b^2)*d^3*e + (7*B*b^2*d*e^3 - (5*B*a*b + 2*A*b^2)*e^4)*x^3 + 3*(7*B*b
^2*d^2*e^2 - (5*B*a*b + 2*A*b^2)*d*e^3)*x^2 + 3*(7*B*b^2*d^3*e - (5*B*a*b + 2*A*b^2)*d^2*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)*sqr
t(b/e) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(15*B*b^2*e^3*x^3 + 105*B*b^2*d^3 - 6*A*a^2*e^3 - 10*(4*B*a*b + 3*A*b^2)
*d^2*e - 2*(2*B*a^2 + 5*A*a*b)*d*e^2 + (161*B*b^2*d*e^2 - 2*(35*B*a*b + 23*A*b^2)*e^3)*x^2 + (245*B*b^2*d^2*e
- 14*(7*B*a*b + 5*A*b^2)*d*e^2 - 2*(5*B*a^2 + 11*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^7*x^3 + 3*d*e^
6*x^2 + 3*d^2*e^5*x + d^3*e^4), 1/30*(15*(7*B*b^2*d^4 - (5*B*a*b + 2*A*b^2)*d^3*e + (7*B*b^2*d*e^3 - (5*B*a*b
+ 2*A*b^2)*e^4)*x^3 + 3*(7*B*b^2*d^2*e^2 - (5*B*a*b + 2*A*b^2)*d*e^3)*x^2 + 3*(7*B*b^2*d^3*e - (5*B*a*b + 2*A*
b^2)*d^2*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*(15*B*b^2*e^3*x^3 + 105*B*b^2*d^3 - 6*A*a^2*e^3 - 10*(4*B*a*b + 3*A*b^2)*d^
2*e - 2*(2*B*a^2 + 5*A*a*b)*d*e^2 + (161*B*b^2*d*e^2 - 2*(35*B*a*b + 23*A*b^2)*e^3)*x^2 + (245*B*b^2*d^2*e - 1
4*(7*B*a*b + 5*A*b^2)*d*e^2 - 2*(5*B*a^2 + 11*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^7*x^3 + 3*d*e^6*x
^2 + 3*d^2*e^5*x + d^3*e^4)]

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giac [B]  time = 3.18, size = 668, normalized size = 2.61 \begin {gather*} \frac {{\left (7 \, B b^{2} d {\left | b \right |} - 5 \, B a b {\left | b \right |} e - 2 \, A b^{2} {\left | b \right |} e\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 )}{\sqrt {b}} + \frac {{\left ({\left ({\left (b x + a\right )} {\left (\frac {15 \, {\left (B b^{9} d^{2} {\left | b \right |} e^{6} - 2 \, B a b^{8} d {\left | b \right |} e^{7} + B a^{2} b^{7} {\left | b \right |} e^{8}\right )} {\left (b x + a\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}} + \frac {23 \, {\left (7 \, B b^{10} d^{3} {\left | b \right |} e^{5} - 19 \, B a b^{9} d^{2} {\left | b \right |} e^{6} - 2 \, A b^{10} d^{2} {\left | b \right |} e^{6} + 17 \, B a^{2} b^{8} d {\left | b \right |} e^{7} + 4 \, A a b^{9} d {\left | b \right |} e^{7} - 5 \, B a^{3} b^{7} {\left | b \right |} e^{8} - 2 \, A a^{2} b^{8} {\left | b \right |} e^{8}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} + \frac {35 \, {\left (7 \, B b^{11} d^{4} {\left | b \right |} e^{4} - 26 \, B a b^{10} d^{3} {\left | b \right |} e^{5} - 2 \, A b^{11} d^{3} {\left | b \right |} e^{5} + 36 \, B a^{2} b^{9} d^{2} {\left | b \right |} e^{6} + 6 \, A a b^{10} d^{2} {\left | b \right |} e^{6} - 22 \, B a^{3} b^{8} d {\left | b \right |} e^{7} - 6 \, A a^{2} b^{9} d {\left | b \right |} e^{7} + 5 \, B a^{4} b^{7} {\left | b \right |} e^{8} + 2 \, A a^{3} b^{8} {\left | b \right |} e^{8}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (7 \, B b^{12} d^{5} {\left | b \right |} e^{3} - 33 \, B a b^{11} d^{4} {\left | b \right |} e^{4} - 2 \, A b^{12} d^{4} {\left | b \right |} e^{4} + 62 \, B a^{2} b^{10} d^{3} {\left | b \right |} e^{5} + 8 \, A a b^{11} d^{3} {\left | b \right |} e^{5} - 58 \, B a^{3} b^{9} d^{2} {\left | b \right |} e^{6} - 12 \, A a^{2} b^{10} d^{2} {\left | b \right |} e^{6} + 27 \, B a^{4} b^{8} d {\left | b \right |} e^{7} + 8 \, A a^{3} b^{9} d {\left | b \right |} e^{7} - 5 \, B a^{5} b^{7} {\left | b \right |} e^{8} - 2 \, A a^{4} b^{8} {\left | b \right |} e^{8}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*B*b^2*d*abs(b) - 5*B*a*b*abs(b)*e - 2*A*b^2*abs(b)*e)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqr
t(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + 1/15*(((b*x + a)*(15*(B*b^9*d^2*abs(b)*e^6 - 2*B*a*b^8*d*abs(b)*e
^7 + B*a^2*b^7*abs(b)*e^8)*(b*x + a)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9) + 23*(7*B*b^10*d^3*abs(b)*e^5
- 19*B*a*b^9*d^2*abs(b)*e^6 - 2*A*b^10*d^2*abs(b)*e^6 + 17*B*a^2*b^8*d*abs(b)*e^7 + 4*A*a*b^9*d*abs(b)*e^7 -
5*B*a^3*b^7*abs(b)*e^8 - 2*A*a^2*b^8*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9)) + 35*(7*B*b^11*d
^4*abs(b)*e^4 - 26*B*a*b^10*d^3*abs(b)*e^5 - 2*A*b^11*d^3*abs(b)*e^5 + 36*B*a^2*b^9*d^2*abs(b)*e^6 + 6*A*a*b^1
0*d^2*abs(b)*e^6 - 22*B*a^3*b^8*d*abs(b)*e^7 - 6*A*a^2*b^9*d*abs(b)*e^7 + 5*B*a^4*b^7*abs(b)*e^8 + 2*A*a^3*b^8
*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9))*(b*x + a) + 15*(7*B*b^12*d^5*abs(b)*e^3 - 33*B*a*b^1
1*d^4*abs(b)*e^4 - 2*A*b^12*d^4*abs(b)*e^4 + 62*B*a^2*b^10*d^3*abs(b)*e^5 + 8*A*a*b^11*d^3*abs(b)*e^5 - 58*B*a
^3*b^9*d^2*abs(b)*e^6 - 12*A*a^2*b^10*d^2*abs(b)*e^6 + 27*B*a^4*b^8*d*abs(b)*e^7 + 8*A*a^3*b^9*d*abs(b)*e^7 -
5*B*a^5*b^7*abs(b)*e^8 - 2*A*a^4*b^8*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9))*sqrt(b*x + a)/(b
^2*d + (b*x + a)*b*e - a*b*e)^(5/2)

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maple [B]  time = 0.03, size = 1092, normalized size = 4.27

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)^(7/2),x)

[Out]

1/30*(75*B*a*b^2*e^4*x^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))-12*((b*x+
a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*a^2*e^3-60*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*b^2*d^2*e+90*A*b^3*d^2*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))+30*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)*B*b^2*e^3*x^3-105*B*b^3*d^4*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
25*B*a*b^2*d*e^3*x^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))+225*B*a*b^2*d
^2*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))-140*((b*x+a)*(e*x+d))^(1/
2)*(b*e)^(1/2)*B*a*b*e^3*x^2+322*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^2*d*e^2*x^2-44*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)*A*a*b*e^3*x+490*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^2*d^2*e*x-80*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)*B*a*b*d^2*e-196*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a*b*d*e^2*x+30*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^3*b^3*e^4+210*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^2*d^3+30
*A*b^3*d^3*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))-140*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)*A*b^2*d*e^2*x-20*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*a*b*d*e^2-105*B*b^3*d*e^3*x^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))-315*B*b^3*d^2*e^2*x^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))-92*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*b^2*e^3*x^2
-315*B*b^3*d^3*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))+75*B*a*b^2*d^3*
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))-20*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)*B*a^2*e^3*x-8*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a^2*d*e^2+90*A*b^3*d*e^3*x^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)))*(b*x+a)^(1/2)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+
d)^(5/2)/e^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)^(5/2)*(B*x+A)/(e*x+d)^(7/2),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 (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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