3.22.6 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac {5 (b d-a e)^2 (-7 a B e+6 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}}+\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e) (-7 a B e+6 A b e+b B d)}{8 b^4}+\frac {5 \sqrt {a+b x} (d+e x)^{3/2} (-7 a B e+6 A b e+b B d)}{12 b^3}+\frac {\sqrt {a+b x} (d+e x)^{5/2} (-7 a B e+6 A b e+b B d)}{3 b^2 (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \]

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Rubi [A]  time = 0.22, antiderivative size = 249, 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, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} (d+e x)^{5/2} (-7 a B e+6 A b e+b B d)}{3 b^2 (b d-a e)}+\frac {5 \sqrt {a+b x} (d+e x)^{3/2} (-7 a B e+6 A b e+b B d)}{12 b^3}+\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e) (-7 a B e+6 A b e+b B d)}{8 b^4}+\frac {5 (b d-a e)^2 (-7 a B e+6 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^{9/2} \sqrt {e}}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(3/2),x]

[Out]

(5*(b*d - a*e)*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^4) + (5*(b*B*d + 6*A*b*e - 7*a*B*
e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^3) + ((b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^
2*(b*d - a*e)) - (2*(A*b - a*B)*(d + e*x)^(7/2))/(b*(b*d - a*e)*Sqrt[a + b*x]) + (5*(b*d - a*e)^2*(b*B*d + 6*A
*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(9/2)*Sqrt[e])

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

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Mathematica [A]  time = 1.66, size = 229, normalized size = 0.92 \begin {gather*} \frac {(d+e x)^{3/2} \left (\frac {(-7 a B e+6 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 {e} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2}}-\frac {48 b^2 (d+e x)^2 (A b-a B) (b d-a e)}{\sqrt {a+b x}}\right )}{24 b^3 (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(3/2),x]

[Out]

((d + e*x)^(3/2)*((-48*b^2*(A*b - a*B)*(b*d - a*e)*(d + e*x)^2)/Sqrt[a + b*x] + ((b*B*d + 6*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[e]*((b*(d +
e*x))/(b*d - a*e))^(3/2))))/(24*b^3*(b*d - a*e)^2)

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IntegrateAlgebraic [A]  time = 0.71, size = 314, normalized size = 1.26 \begin {gather*} \frac {5 (b d-a e)^2 (-7 a B e+6 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{9/2} \sqrt {e}}-\frac {\sqrt {d+e x} (b d-a e)^2 \left (\frac {48 A b^4 (d+e x)^3}{(a+b x)^3}-\frac {198 A b^3 e (d+e x)^2}{(a+b x)^2}+\frac {240 A b^2 e^2 (d+e x)}{a+b x}-\frac {48 a b^3 B (d+e x)^3}{(a+b x)^3}-\frac {33 b^3 B d (d+e x)^2}{(a+b x)^2}+\frac {231 a b^2 B e (d+e x)^2}{(a+b x)^2}+\frac {40 b^2 B d e (d+e x)}{a+b x}-\frac {280 a b B e^2 (d+e x)}{a+b x}+105 a B e^3-90 A b e^3-15 b B d e^2\right )}{24 b^4 \sqrt {a+b x} \left (\frac {b (d+e x)}{a+b x}-e\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(3/2),x]

[Out]

-1/24*((b*d - a*e)^2*Sqrt[d + e*x]*(-15*b*B*d*e^2 - 90*A*b*e^3 + 105*a*B*e^3 + (40*b^2*B*d*e*(d + e*x))/(a + b
*x) + (240*A*b^2*e^2*(d + e*x))/(a + b*x) - (280*a*b*B*e^2*(d + e*x))/(a + b*x) - (33*b^3*B*d*(d + e*x)^2)/(a
+ b*x)^2 - (198*A*b^3*e*(d + e*x)^2)/(a + b*x)^2 + (231*a*b^2*B*e*(d + e*x)^2)/(a + b*x)^2 + (48*A*b^4*(d + e*
x)^3)/(a + b*x)^3 - (48*a*b^3*B*(d + e*x)^3)/(a + b*x)^3))/(b^4*Sqrt[a + b*x]*(-e + (b*(d + e*x))/(a + b*x))^3
) + (5*(b*d - a*e)^2*(b*B*d + 6*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(8*
b^(9/2)*Sqrt[e])

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fricas [B]  time = 2.73, size = 872, normalized size = 3.50 \begin {gather*} \left [-\frac {15 \, {\left (B a b^{3} d^{3} - 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e + 3 \, {\left (5 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} d e^{2} - {\left (7 \, B a^{4} - 6 \, A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{3} - 3 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d e^{2} - {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3}\right )} x\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 (8 \, B b^{4} e^{3} x^{3} + 3 \, {\left (27 \, B a b^{3} - 16 \, A b^{4}\right )} d^{2} e - 10 \, {\left (19 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3} + 2 \, {\left (13 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} e^{3}\right )} x^{2} + {\left (33 \, B b^{4} d^{2} e - 2 \, {\left (34 \, B a b^{3} - 27 \, A b^{4}\right )} d e^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, {\left (b^{6} e x + a b^{5} e\right )}}, -\frac {15 \, {\left (B a b^{3} d^{3} - 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e + 3 \, {\left (5 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} d e^{2} - {\left (7 \, B a^{4} - 6 \, A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{3} - 3 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d e^{2} - {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3}\right )} x\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 (8 \, B b^{4} e^{3} x^{3} + 3 \, {\left (27 \, B a b^{3} - 16 \, A b^{4}\right )} d^{2} e - 10 \, {\left (19 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} e^{3} + 2 \, {\left (13 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} e^{3}\right )} x^{2} + {\left (33 \, B b^{4} d^{2} e - 2 \, {\left (34 \, B a b^{3} - 27 \, A b^{4}\right )} d e^{2} + 5 \, {\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (b^{6} e x + a b^{5} e\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)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(15*(B*a*b^3*d^3 - 3*(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e + 3*(5*B*a^3*b - 4*A*a^2*b^2)*d*e^2 - (7*B*a^4 - 6
*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a*b^3 - 2*A*b^4)*d^2*e + 3*(5*B*a^2*b^2 - 4*A*a*b^3)*d*e^2 - (7*B*a^3*b -
6*A*a^2*b^2)*e^3)*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*x + b*d + a*e)*sqr
t(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(8*B*b^4*e^3*x^3 + 3*(27*B*a*b^3 - 16*A*b^4)
*d^2*e - 10*(19*B*a^2*b^2 - 15*A*a*b^3)*d*e^2 + 15*(7*B*a^3*b - 6*A*a^2*b^2)*e^3 + 2*(13*B*b^4*d*e^2 - (7*B*a*
b^3 - 6*A*b^4)*e^3)*x^2 + (33*B*b^4*d^2*e - 2*(34*B*a*b^3 - 27*A*b^4)*d*e^2 + 5*(7*B*a^2*b^2 - 6*A*a*b^3)*e^3)
*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*e*x + a*b^5*e), -1/48*(15*(B*a*b^3*d^3 - 3*(3*B*a^2*b^2 - 2*A*a*b^3)*d^2
*e + 3*(5*B*a^3*b - 4*A*a^2*b^2)*d*e^2 - (7*B*a^4 - 6*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a*b^3 - 2*A*b^4)*d^2*
e + 3*(5*B*a^2*b^2 - 4*A*a*b^3)*d*e^2 - (7*B*a^3*b - 6*A*a^2*b^2)*e^3)*x)*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*(8*B*b^4*e
^3*x^3 + 3*(27*B*a*b^3 - 16*A*b^4)*d^2*e - 10*(19*B*a^2*b^2 - 15*A*a*b^3)*d*e^2 + 15*(7*B*a^3*b - 6*A*a^2*b^2)
*e^3 + 2*(13*B*b^4*d*e^2 - (7*B*a*b^3 - 6*A*b^4)*e^3)*x^2 + (33*B*b^4*d^2*e - 2*(34*B*a*b^3 - 27*A*b^4)*d*e^2
+ 5*(7*B*a^2*b^2 - 6*A*a*b^3)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*e*x + a*b^5*e)]

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giac [B]  time = 2.99, size = 477, normalized size = 1.92 \begin {gather*} \frac {1}{24} \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |} e^{2}}{b^{6}} + \frac {{\left (13 \, B b^{18} d {\left | b \right |} e^{5} - 19 \, B a b^{17} {\left | b \right |} e^{6} + 6 \, A b^{18} {\left | b \right |} e^{6}\right )} e^{\left (-4\right )}}{b^{23}}\right )} + \frac {3 \, {\left (11 \, B b^{19} d^{2} {\left | b \right |} e^{4} - 40 \, B a b^{18} d {\left | b \right |} e^{5} + 18 \, A b^{19} d {\left | b \right |} e^{5} + 29 \, B a^{2} b^{17} {\left | b \right |} e^{6} - 18 \, A a b^{18} {\left | b \right |} e^{6}\right )} e^{\left (-4\right )}}{b^{23}}\right )} - \frac {5 \, {\left (B b^{\frac {7}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - 9 \, B a b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 6 \, A b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 15 \, B a^{2} b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 12 \, A a b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 7 \, B a^{3} \sqrt {b} {\left | b \right |} e^{\frac {7}{2}} + 6 \, A a^{2} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {7}{2}}\right )} e^{\left (-1\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 )}^{2}\right )}{16 \, b^{6}} + \frac {4 \, {\left (B a b^{\frac {7}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - A b^{\frac {9}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - 3 \, B a^{2} b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 3 \, A a b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + 3 \, B a^{3} b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 3 \, A a^{2} b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {5}{2}} - B a^{4} \sqrt {b} {\left | b \right |} e^{\frac {7}{2}} + A a^{3} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {7}{2}}\right )}}{{\left (b^{2} d - a b e - {\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 )}^{2}\right )} b^{5}} \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)^(3/2),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^2/b^6 + (13*B*b^18
*d*abs(b)*e^5 - 19*B*a*b^17*abs(b)*e^6 + 6*A*b^18*abs(b)*e^6)*e^(-4)/b^23) + 3*(11*B*b^19*d^2*abs(b)*e^4 - 40*
B*a*b^18*d*abs(b)*e^5 + 18*A*b^19*d*abs(b)*e^5 + 29*B*a^2*b^17*abs(b)*e^6 - 18*A*a*b^18*abs(b)*e^6)*e^(-4)/b^2
3) - 5/16*(B*b^(7/2)*d^3*abs(b)*e^(1/2) - 9*B*a*b^(5/2)*d^2*abs(b)*e^(3/2) + 6*A*b^(7/2)*d^2*abs(b)*e^(3/2) +
15*B*a^2*b^(3/2)*d*abs(b)*e^(5/2) - 12*A*a*b^(5/2)*d*abs(b)*e^(5/2) - 7*B*a^3*sqrt(b)*abs(b)*e^(7/2) + 6*A*a^2
*b^(3/2)*abs(b)*e^(7/2))*e^(-1)*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*(B*a*b^(7/2)*d^3*abs(b)*e^(1/2) - A*b^(9/2)*d^3*abs(b)*e^(1/2) - 3*B*a^2*b^(5/2)*d^2*abs(b)*e^(3/2) + 3
*A*a*b^(7/2)*d^2*abs(b)*e^(3/2) + 3*B*a^3*b^(3/2)*d*abs(b)*e^(5/2) - 3*A*a^2*b^(5/2)*d*abs(b)*e^(5/2) - B*a^4*
sqrt(b)*abs(b)*e^(7/2) + A*a^3*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)*b^5)

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maple [B]  time = 0.03, size = 1184, normalized size = 4.76

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)^(3/2),x)

[Out]

1/48*(e*x+d)^(1/2)*(16*B*x^3*b^3*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*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*e^3-105*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^4*e^3-136*B*x*a*b^2*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+15*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^3-96*A*b^3*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e
)^(1/2)+210*B*a^3*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+15*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^3+66*B*x*b^3*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*A*a^2*b*e^2*((b*
x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+162*B*a*b^2*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*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^2*e^3+90*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*b^4*d^2*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))*x*a^3*b*e^3-180*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^2*d*e^2+90*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*b^3*d
^2*e+225*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*d*e^2-135*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^2*d^2*e+24*A*x^2*b^3*e^2*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)-380*B*a^2*b*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*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^3*d*e^2+225*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^2*d*e^2-135*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^2*e-28*B*x^2*a*b^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+52*B*x^2*b^3*d*
e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-60*A*x*a*b^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+108*A*x*b^3*d*e*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+70*B*x*a^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+300*A*a*b^2*d*e*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(1/2)/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)^(3/2),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 )}^{3/2}} \,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)^(3/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \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)**(3/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**(5/2)/(a + b*x)**(3/2), x)

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