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

Optimal. Leaf size=246 \[ -\frac {5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}-\frac {5 \sqrt {a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac {\sqrt {a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e} \]

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Rubi [A]  time = 0.21, antiderivative size = 246, 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 = {80, 50, 63, 217, 206} \begin {gather*} -\frac {5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac {\sqrt {a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}-\frac {5 \sqrt {a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*d - a*e)^2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(64*b^4*e) - (5*(b*d - a*e)*(b*B*d
- 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(96*b^3*e) - ((b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*(d
+ e*x)^(5/2))/(24*b^2*e) + (B*Sqrt[a + b*x]*(d + e*x)^(7/2))/(4*b*e) - (5*(b*d - a*e)^3*(b*B*d - 8*A*b*e + 7*
a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b^(9/2)*e^(3/2))

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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}}{\sqrt {a+b x}} \, dx &=\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {b d}{2}+\frac {7 a e}{2}\right )\right ) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}} \, dx}{4 b e}\\ &=-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {(5 (b d-a e) (b B d-8 A b e+7 a B e)) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^2 e}\\ &=-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^2 (b B d-8 A b e+7 a B e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{64 b^3 e}\\ &=-\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^3 (b B d-8 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b^4 e}\\ &=-\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^3 (b B d-8 A b e+7 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 )}{64 b^5 e}\\ &=-\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^3 (b B d-8 A b e+7 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 )}{64 b^5 e}\\ &=-\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {5 (b d-a e)^3 (b B d-8 A b e+7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 1.58, size = 210, normalized size = 0.85 \begin {gather*} \frac {\sqrt {d+e x} \left (48 b^3 B \sqrt {e} \sqrt {a+b x} (d+e x)^3-\frac {(7 a B e-8 A b e+b B d) \left (\sqrt {e} \sqrt {a+b x} \sqrt {\frac {b (d+e x)}{b d-a e}} \left (15 a^2 e^2-10 a b e (4 d+e x)+b^2 \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )+15 (b d-a e)^{5/2} \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )}{\sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{192 b^4 e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(48*b^3*B*Sqrt[e]*Sqrt[a + b*x]*(d + e*x)^3 - ((b*B*d - 8*A*b*e + 7*a*B*e)*(Sqrt[e]*Sqrt[a + b*
x]*Sqrt[(b*(d + e*x))/(b*d - a*e)]*(15*a^2*e^2 - 10*a*b*e*(4*d + e*x) + b^2*(33*d^2 + 26*d*e*x + 8*e^2*x^2)) +
15*(b*d - a*e)^(5/2)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/Sqrt[(b*(d + e*x))/(b*d - a*e)]))/(19
2*b^4*e^(3/2))

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IntegrateAlgebraic [A]  time = 0.45, size = 346, normalized size = 1.41 \begin {gather*} \frac {\sqrt {a+b x} (b d-a e)^3 \left (-\frac {584 A b^3 e^2 (a+b x)}{d+e x}+\frac {440 A b^2 e^3 (a+b x)^2}{(d+e x)^2}-\frac {120 A b e^4 (a+b x)^3}{(d+e x)^3}+\frac {73 b^3 B d e (a+b x)}{d+e x}-279 a b^3 B e+\frac {511 a b^2 B e^2 (a+b x)}{d+e x}-\frac {55 b^2 B d e^2 (a+b x)^2}{(d+e x)^2}+\frac {105 a B e^4 (a+b x)^3}{(d+e x)^3}-\frac {385 a b B e^3 (a+b x)^2}{(d+e x)^2}+\frac {15 b B d e^3 (a+b x)^3}{(d+e x)^3}+264 A b^4 e+15 b^4 B d\right )}{192 b^4 e \sqrt {d+e x} \left (b-\frac {e (a+b x)}{d+e x}\right )^4}-\frac {5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^3*Sqrt[a + b*x]*(15*b^4*B*d + 264*A*b^4*e - 279*a*b^3*B*e + (15*b*B*d*e^3*(a + b*x)^3)/(d + e*x)^
3 - (120*A*b*e^4*(a + b*x)^3)/(d + e*x)^3 + (105*a*B*e^4*(a + b*x)^3)/(d + e*x)^3 - (55*b^2*B*d*e^2*(a + b*x)^
2)/(d + e*x)^2 + (440*A*b^2*e^3*(a + b*x)^2)/(d + e*x)^2 - (385*a*b*B*e^3*(a + b*x)^2)/(d + e*x)^2 + (73*b^3*B
*d*e*(a + b*x))/(d + e*x) - (584*A*b^3*e^2*(a + b*x))/(d + e*x) + (511*a*b^2*B*e^2*(a + b*x))/(d + e*x)))/(192
*b^4*e*Sqrt[d + e*x]*(b - (e*(a + b*x))/(d + e*x))^4) - (5*(b*d - a*e)^3*(b*B*d - 8*A*b*e + 7*a*B*e)*ArcTanh[(
Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(64*b^(9/2)*e^(3/2))

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fricas [A]  time = 1.68, size = 772, normalized size = 3.14 \begin {gather*} \left [-\frac {15 \, {\left (B b^{4} d^{4} + 4 \, {\left (B a b^{3} - 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (5 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {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 x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (191 \, B a b^{3} - 264 \, A b^{4}\right )} d^{2} e^{2} + 5 \, {\left (53 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} - 15 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (17 \, B b^{4} d e^{3} - {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (59 \, B b^{4} d^{2} e^{2} - 2 \, {\left (43 \, B a b^{3} - 52 \, A b^{4}\right )} d e^{3} + 5 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{768 \, b^{5} e^{2}}, \frac {15 \, {\left (B b^{4} d^{4} + 4 \, {\left (B a b^{3} - 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (5 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (191 \, B a b^{3} - 264 \, A b^{4}\right )} d^{2} e^{2} + 5 \, {\left (53 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} - 15 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (17 \, B b^{4} d e^{3} - {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (59 \, B b^{4} d^{2} e^{2} - 2 \, {\left (43 \, B a b^{3} - 52 \, A b^{4}\right )} d e^{3} + 5 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{384 \, b^{5} e^{2}}\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)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(15*(B*b^4*d^4 + 4*(B*a*b^3 - 2*A*b^4)*d^3*e - 6*(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 4*(5*B*a^3*b - 6*
A*a^2*b^2)*d*e^3 - (7*B*a^4 - 8*A*a^3*b)*e^4)*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*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(48*B*b^4*e^4*x^3 +
15*B*b^4*d^3*e - (191*B*a*b^3 - 264*A*b^4)*d^2*e^2 + 5*(53*B*a^2*b^2 - 64*A*a*b^3)*d*e^3 - 15*(7*B*a^3*b - 8*
A*a^2*b^2)*e^4 + 8*(17*B*b^4*d*e^3 - (7*B*a*b^3 - 8*A*b^4)*e^4)*x^2 + 2*(59*B*b^4*d^2*e^2 - 2*(43*B*a*b^3 - 52
*A*b^4)*d*e^3 + 5*(7*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^2), 1/384*(15*(B*b^4*d
^4 + 4*(B*a*b^3 - 2*A*b^4)*d^3*e - 6*(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 4*(5*B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (
7*B*a^4 - 8*A*a^3*b)*e^4)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(
b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(48*B*b^4*e^4*x^3 + 15*B*b^4*d^3*e - (191*B*a*b^3 - 264*A*
b^4)*d^2*e^2 + 5*(53*B*a^2*b^2 - 64*A*a*b^3)*d*e^3 - 15*(7*B*a^3*b - 8*A*a^2*b^2)*e^4 + 8*(17*B*b^4*d*e^3 - (7
*B*a*b^3 - 8*A*b^4)*e^4)*x^2 + 2*(59*B*b^4*d^2*e^2 - 2*(43*B*a*b^3 - 52*A*b^4)*d*e^3 + 5*(7*B*a^2*b^2 - 8*A*a*
b^3)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^2)]

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giac [B]  time = 1.94, size = 1055, normalized size = 4.29

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)^(1/2),x, algorithm="giac")

[Out]

-1/192*(192*((b^2*d - a*b*e)*e^(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*
b*e)))/sqrt(b) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*A*d^2*abs(b)/b^2 - 16*(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^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7
*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)
*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*d*abs(b)*e
/b^2 - 48*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d
+ (b*x + a)*b*e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2) + 2
*a)*sqrt(b*x + a))*B*d^2*abs(b)/b^3 - 8*(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^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/
b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) +
sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*A*abs(b)*e^2/b^2 - (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*
x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12
*d*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^
11*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^
4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*abs(
b)*e^2/b^2 - 96*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(
b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-
2) + 2*a)*sqrt(b*x + a))*A*d*abs(b)*e/b^3)/b

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maple [B]  time = 0.03, size = 968, normalized size = 3.93 \begin {gather*} -\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (120 A \,a^{3} b \,e^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-360 A \,a^{2} b^{2} d \,e^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+360 A a \,b^{3} d^{2} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-120 A \,b^{4} d^{3} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-105 B \,a^{4} e^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+300 B \,a^{3} b d \,e^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-270 B \,a^{2} b^{2} d^{2} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+60 B a \,b^{3} d^{3} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+15 B \,b^{4} d^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-96 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} e^{3} x^{3}-128 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,b^{3} e^{3} x^{2}+112 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B a \,b^{2} e^{3} x^{2}-272 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} d \,e^{2} x^{2}+160 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A a \,b^{2} e^{3} x -416 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,b^{3} d \,e^{2} x -140 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,a^{2} b \,e^{3} x +344 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B a \,b^{2} d \,e^{2} x -236 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} d^{2} e x -240 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,a^{2} b \,e^{3}+640 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A a \,b^{2} d \,e^{2}-528 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,b^{3} d^{2} e +210 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,a^{3} e^{3}-530 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,a^{2} b d \,e^{2}+382 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B a \,b^{2} d^{2} e -30 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} d^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{4} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(1/2),x)

[Out]

-1/384*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(-360*A*a^2*b^2*d*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))-30*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*B*b^3*d^3-105*B*a^4*e^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))+15*B*b^4*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))+120*A*a^3*b*e^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))-120*A*b^4*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))+210*(
b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*B*a^3*e^3+344*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*B*a*b^2*d*e^2*x+360*A*a*b
^3*d^2*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))-96*(b*e)^(1/2)*((b*x+a)
*(e*x+d))^(1/2)*B*b^3*e^3*x^3-128*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*A*b^3*e^3*x^2+60*B*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))-240*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*A*a
^2*b*e^3-528*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*A*b^3*d^2*e-270*B*a^2*b^2*d^2*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))+300*B*a^3*b*d*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))+160*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*A*a*b^2*e^3*x-416*(b*e)^(1/2)*((b*x+a
)*(e*x+d))^(1/2)*A*b^3*d*e^2*x-140*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*B*a^2*b*e^3*x-236*(b*e)^(1/2)*((b*x+a)*
(e*x+d))^(1/2)*B*b^3*d^2*e*x+112*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*B*a*b^2*e^3*x^2-272*(b*e)^(1/2)*((b*x+a)*
(e*x+d))^(1/2)*B*b^3*d*e^2*x^2+640*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*A*a*b^2*d*e^2-530*(b*e)^(1/2)*((b*x+a)*
(e*x+d))^(1/2)*B*a^2*b*d*e^2+382*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*B*a*b^2*d^2*e)/b^4/e/((b*x+a)*(e*x+d))^(1
/2)/(b*e)^(1/2)

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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)^(1/2),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}}{\sqrt {a+b\,x}} \,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)^(1/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(1/2), 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)**(1/2),x)

[Out]

Timed out

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