3.17.19 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=274 \[ -\frac {7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}}+\frac {7 e \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}+\frac {7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac {7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

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Rubi [A]  time = 0.24, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 50, 63, 208} \begin {gather*} -\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}+\frac {7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}+\frac {7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac {7 e \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}-\frac {7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}}-\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(4*b^5) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d
+ e*x)^(3/2))/(12*b^4) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)) - ((4*b*B*d
+ 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(9/2))/(2*b*(b*d
- a*e)*(a + b*x)^2) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqr
t[b*d - a*e]])/(4*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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^3} \, dx &=-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(4 b B d+5 A b e-9 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{4 b (b d-a e)}\\ &=-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (4 b B d+5 A b e-9 a B e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{8 b^2 (b d-a e)}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (4 b B d+5 A b e-9 a B e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{8 b^3}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (b d-a e) (4 b B d+5 A b e-9 a B e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{8 b^4}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {\left (7 e (b d-a e)^2 (4 b B d+5 A b e-9 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 b^5}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {\left (7 (b d-a e)^2 (4 b B d+5 A b e-9 a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}-\frac {7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 97, normalized size = 0.35 \begin {gather*} \frac {(d+e x)^{9/2} \left (\frac {e (-9 a B e+5 A b e+4 b B d) \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac {9 a B-9 A b}{(a+b x)^2}\right )}{18 b (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x]

[Out]

((d + e*x)^(9/2)*((-9*A*b + 9*a*B)/(a + b*x)^2 + (e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Hypergeometric2F1[2, 9/2, 11
/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^2))/(18*b*(b*d - a*e))

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IntegrateAlgebraic [A]  time = 1.14, size = 477, normalized size = 1.74 \begin {gather*} \frac {e \sqrt {d+e x} \left (945 a^4 B e^4-525 a^3 A b e^4+1575 a^3 b B e^3 (d+e x)-3255 a^3 b B d e^3-875 a^2 A b^2 e^3 (d+e x)+1575 a^2 A b^2 d e^3+4095 a^2 b^2 B d^2 e^2+504 a^2 b^2 B e^2 (d+e x)^2-3850 a^2 b^2 B d e^2 (d+e x)-1575 a A b^3 d^2 e^2-280 a A b^3 e^2 (d+e x)^2+1750 a A b^3 d e^2 (d+e x)-2205 a b^3 B d^3 e+2975 a b^3 B d^2 e (d+e x)-72 a b^3 B e (d+e x)^3-728 a b^3 B d e (d+e x)^2+525 A b^4 d^3 e-875 A b^4 d^2 e (d+e x)+40 A b^4 e (d+e x)^3+280 A b^4 d e (d+e x)^2+420 b^4 B d^4-700 b^4 B d^3 (d+e x)+224 b^4 B d^2 (d+e x)^2+24 b^4 B (d+e x)^4+32 b^4 B d (d+e x)^3\right )}{60 b^5 (a e+b (d+e x)-b d)^2}-\frac {7 (b d-a e)^2 \left (-9 a B e^2+5 A b e^2+4 b B d e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 b^{11/2} \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x]

[Out]

(e*Sqrt[d + e*x]*(420*b^4*B*d^4 + 525*A*b^4*d^3*e - 2205*a*b^3*B*d^3*e - 1575*a*A*b^3*d^2*e^2 + 4095*a^2*b^2*B
*d^2*e^2 + 1575*a^2*A*b^2*d*e^3 - 3255*a^3*b*B*d*e^3 - 525*a^3*A*b*e^4 + 945*a^4*B*e^4 - 700*b^4*B*d^3*(d + e*
x) - 875*A*b^4*d^2*e*(d + e*x) + 2975*a*b^3*B*d^2*e*(d + e*x) + 1750*a*A*b^3*d*e^2*(d + e*x) - 3850*a^2*b^2*B*
d*e^2*(d + e*x) - 875*a^2*A*b^2*e^3*(d + e*x) + 1575*a^3*b*B*e^3*(d + e*x) + 224*b^4*B*d^2*(d + e*x)^2 + 280*A
*b^4*d*e*(d + e*x)^2 - 728*a*b^3*B*d*e*(d + e*x)^2 - 280*a*A*b^3*e^2*(d + e*x)^2 + 504*a^2*b^2*B*e^2*(d + e*x)
^2 + 32*b^4*B*d*(d + e*x)^3 + 40*A*b^4*e*(d + e*x)^3 - 72*a*b^3*B*e*(d + e*x)^3 + 24*b^4*B*(d + e*x)^4))/(60*b
^5*(-(b*d) + a*e + b*(d + e*x))^2) - (7*(b*d - a*e)^2*(4*b*B*d*e + 5*A*b*e^2 - 9*a*B*e^2)*ArcTan[(Sqrt[b]*Sqrt
[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(4*b^(11/2)*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 1.68, size = 1060, normalized size = 3.87

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)^3,x, algorithm="fricas")

[Out]

[1/120*(105*(4*B*a^2*b^2*d^2*e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*e
 - (13*B*a*b^3 - 5*A*b^4)*d*e^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*
A*a*b^3)*d*e^2 + (9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x +
 d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(24*B*b^4*e^3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15
*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (
9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e - 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b
^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 195*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*
(9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/60*(105*(4*B*a^2*b^2*d^2*
e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*e - (13*B*a*b^3 - 5*A*b^4)*d*e
^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*A*a*b^3)*d*e^2 + (9*B*a^3*b -
 5*A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (24*B*b
^4*e^3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*a^2*b^2)*
d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e
 - 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 19
5*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*(9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/
(b^7*x^2 + 2*a*b^6*x + a^2*b^5)]

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giac [B]  time = 1.39, size = 607, normalized size = 2.22 \begin {gather*} \frac {7 \, {\left (4 \, B b^{3} d^{3} e - 17 \, B a b^{2} d^{2} e^{2} + 5 \, A b^{3} d^{2} e^{2} + 22 \, B a^{2} b d e^{3} - 10 \, A a b^{2} d e^{3} - 9 \, B a^{3} e^{4} + 5 \, A a^{2} b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e - 4 \, \sqrt {x e + d} B b^{4} d^{4} e - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{2} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{2} + 27 \, \sqrt {x e + d} B a b^{3} d^{3} e^{2} - 11 \, \sqrt {x e + d} A b^{4} d^{3} e^{2} + 38 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{3} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{3} - 57 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{3} + 33 \, \sqrt {x e + d} A a b^{3} d^{2} e^{3} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{4} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{4} + 49 \, \sqrt {x e + d} B a^{3} b d e^{4} - 33 \, \sqrt {x e + d} A a^{2} b^{2} d e^{4} - 15 \, \sqrt {x e + d} B a^{4} e^{5} + 11 \, \sqrt {x e + d} A a^{3} b e^{5}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{12} e + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{12} d e + 45 \, \sqrt {x e + d} B b^{12} d^{2} e - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{11} e^{2} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{12} e^{2} - 135 \, \sqrt {x e + d} B a b^{11} d e^{2} + 45 \, \sqrt {x e + d} A b^{12} d e^{2} + 90 \, \sqrt {x e + d} B a^{2} b^{10} e^{3} - 45 \, \sqrt {x e + d} A a b^{11} e^{3}\right )}}{15 \, b^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

7/4*(4*B*b^3*d^3*e - 17*B*a*b^2*d^2*e^2 + 5*A*b^3*d^2*e^2 + 22*B*a^2*b*d*e^3 - 10*A*a*b^2*d*e^3 - 9*B*a^3*e^4
+ 5*A*a^2*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - 1/4*(4*(x*e + d)^(3
/2)*B*b^4*d^3*e - 4*sqrt(x*e + d)*B*b^4*d^4*e - 25*(x*e + d)^(3/2)*B*a*b^3*d^2*e^2 + 13*(x*e + d)^(3/2)*A*b^4*
d^2*e^2 + 27*sqrt(x*e + d)*B*a*b^3*d^3*e^2 - 11*sqrt(x*e + d)*A*b^4*d^3*e^2 + 38*(x*e + d)^(3/2)*B*a^2*b^2*d*e
^3 - 26*(x*e + d)^(3/2)*A*a*b^3*d*e^3 - 57*sqrt(x*e + d)*B*a^2*b^2*d^2*e^3 + 33*sqrt(x*e + d)*A*a*b^3*d^2*e^3
- 17*(x*e + d)^(3/2)*B*a^3*b*e^4 + 13*(x*e + d)^(3/2)*A*a^2*b^2*e^4 + 49*sqrt(x*e + d)*B*a^3*b*d*e^4 - 33*sqrt
(x*e + d)*A*a^2*b^2*d*e^4 - 15*sqrt(x*e + d)*B*a^4*e^5 + 11*sqrt(x*e + d)*A*a^3*b*e^5)/(((x*e + d)*b - b*d + a
*e)^2*b^5) + 2/15*(3*(x*e + d)^(5/2)*B*b^12*e + 10*(x*e + d)^(3/2)*B*b^12*d*e + 45*sqrt(x*e + d)*B*b^12*d^2*e
- 15*(x*e + d)^(3/2)*B*a*b^11*e^2 + 5*(x*e + d)^(3/2)*A*b^12*e^2 - 135*sqrt(x*e + d)*B*a*b^11*d*e^2 + 45*sqrt(
x*e + d)*A*b^12*d*e^2 + 90*sqrt(x*e + d)*B*a^2*b^10*e^3 - 45*sqrt(x*e + d)*A*a*b^11*e^3)/b^15

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maple [B]  time = 0.03, size = 940, normalized size = 3.43 \begin {gather*} -\frac {11 \sqrt {e x +d}\, A \,a^{3} e^{5}}{4 \left (b x e +a e \right )^{2} b^{4}}+\frac {33 \sqrt {e x +d}\, A \,a^{2} d \,e^{4}}{4 \left (b x e +a e \right )^{2} b^{3}}-\frac {33 \sqrt {e x +d}\, A a \,d^{2} e^{3}}{4 \left (b x e +a e \right )^{2} b^{2}}+\frac {11 \sqrt {e x +d}\, A \,d^{3} e^{2}}{4 \left (b x e +a e \right )^{2} b}+\frac {15 \sqrt {e x +d}\, B \,a^{4} e^{5}}{4 \left (b x e +a e \right )^{2} b^{5}}-\frac {49 \sqrt {e x +d}\, B \,a^{3} d \,e^{4}}{4 \left (b x e +a e \right )^{2} b^{4}}+\frac {57 \sqrt {e x +d}\, B \,a^{2} d^{2} e^{3}}{4 \left (b x e +a e \right )^{2} b^{3}}-\frac {27 \sqrt {e x +d}\, B a \,d^{3} e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}+\frac {\sqrt {e x +d}\, B \,d^{4} e}{\left (b x e +a e \right )^{2} b}-\frac {13 \left (e x +d \right )^{\frac {3}{2}} A \,a^{2} e^{4}}{4 \left (b x e +a e \right )^{2} b^{3}}+\frac {35 A \,a^{2} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {13 \left (e x +d \right )^{\frac {3}{2}} A a d \,e^{3}}{2 \left (b x e +a e \right )^{2} b^{2}}-\frac {35 A a d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {13 \left (e x +d \right )^{\frac {3}{2}} A \,d^{2} e^{2}}{4 \left (b x e +a e \right )^{2} b}+\frac {35 A \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} B \,a^{3} e^{4}}{4 \left (b x e +a e \right )^{2} b^{4}}-\frac {63 B \,a^{3} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{5}}-\frac {19 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} d \,e^{3}}{2 \left (b x e +a e \right )^{2} b^{3}}+\frac {77 B \,a^{2} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {25 \left (e x +d \right )^{\frac {3}{2}} B a \,d^{2} e^{2}}{4 \left (b x e +a e \right )^{2} b^{2}}-\frac {119 B a \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {\left (e x +d \right )^{\frac {3}{2}} B \,d^{3} e}{\left (b x e +a e \right )^{2} b}+\frac {7 B \,d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {6 \sqrt {e x +d}\, A a \,e^{3}}{b^{4}}+\frac {6 \sqrt {e x +d}\, A d \,e^{2}}{b^{3}}+\frac {12 \sqrt {e x +d}\, B \,a^{2} e^{3}}{b^{5}}-\frac {18 \sqrt {e x +d}\, B a d \,e^{2}}{b^{4}}+\frac {6 \sqrt {e x +d}\, B \,d^{2} e}{b^{3}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} A \,e^{2}}{3 b^{3}}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B a \,e^{2}}{b^{4}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} B d e}{3 b^{3}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} B e}{5 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^3,x)

[Out]

2/3/b^3*A*(e*x+d)^(3/2)*e^2+2/5*e/b^3*B*(e*x+d)^(5/2)+4/3*e/b^3*B*(e*x+d)^(3/2)*d-63/4/b^5/((a*e-b*d)*b)^(1/2)
*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a^3*e^4+25/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a*d^2*e^2+33/4/b
^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a^2*d*e^4-33/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a*d^2*e^3-49/4/b^4/(b*e*x+a*
e)^2*(e*x+d)^(1/2)*B*a^3*d*e^4+57/4/b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*d^2*e^3-119/4/b^3/((a*e-b*d)*b)^(1/2
)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a*d^2*e^2-27/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a*d^3*e^2-35/
2/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*a*d*e^3+77/2/b^4/((a*e-b*d)*b)^(1/2)*a
rctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a^2*d*e^3-13/4/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*d^2*e^2+17/4/b^4/(
b*e*x+a*e)^2*(e*x+d)^(3/2)*a^3*e^4*B-11/4/b^4/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a^3*e^5+11/4/b/(b*e*x+a*e)^2*(e*x+
d)^(1/2)*A*d^3*e^2+15/4/b^5/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^4*e^5+35/4/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^
(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*a^2*e^4+35/4/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b
)*A*d^2*e^2+e/b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*d^4+7*e/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*
b)^(1/2)*b)*B*d^3-e/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*d^3-18/b^4*B*(e*x+d)^(1/2)*a*d*e^2-13/4/b^3/(b*e*x+a*e)^2*
(e*x+d)^(3/2)*A*a^2*e^4+13/2/b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*a*d*e^3-19/2/b^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*
a^2*d*e^3+6*e/b^3*B*(e*x+d)^(1/2)*d^2-2/b^4*B*(e*x+d)^(3/2)*a*e^2+12/b^5*B*(e*x+d)^(1/2)*a^2*e^3-6/b^4*A*(e*x+
d)^(1/2)*a*e^3+6/b^3*A*(e*x+d)^(1/2)*d*e^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)^(7/2)/(b*x+a)^3,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 positive or negative?

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mupad [B]  time = 1.33, size = 561, normalized size = 2.05 \begin {gather*} \left (\frac {2\,A\,e^2-2\,B\,d\,e}{3\,b^3}+\frac {2\,B\,e\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{3\,b^6}\right )\,{\left (d+e\,x\right )}^{3/2}+\left (\frac {\left (\frac {2\,A\,e^2-2\,B\,d\,e}{b^3}+\frac {2\,B\,e\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{b^6}\right )\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{b^3}-\frac {6\,B\,e\,{\left (a\,e-b\,d\right )}^2}{b^5}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {17\,B\,a^3\,b\,e^4}{4}+\frac {19\,B\,a^2\,b^2\,d\,e^3}{2}+\frac {13\,A\,a^2\,b^2\,e^4}{4}-\frac {25\,B\,a\,b^3\,d^2\,e^2}{4}-\frac {13\,A\,a\,b^3\,d\,e^3}{2}+B\,b^4\,d^3\,e+\frac {13\,A\,b^4\,d^2\,e^2}{4}\right )-\sqrt {d+e\,x}\,\left (\frac {15\,B\,a^4\,e^5}{4}-\frac {49\,B\,a^3\,b\,d\,e^4}{4}-\frac {11\,A\,a^3\,b\,e^5}{4}+\frac {57\,B\,a^2\,b^2\,d^2\,e^3}{4}+\frac {33\,A\,a^2\,b^2\,d\,e^4}{4}-\frac {27\,B\,a\,b^3\,d^3\,e^2}{4}-\frac {33\,A\,a\,b^3\,d^2\,e^3}{4}+B\,b^4\,d^4\,e+\frac {11\,A\,b^4\,d^3\,e^2}{4}\right )}{b^7\,{\left (d+e\,x\right )}^2-\left (2\,b^7\,d-2\,a\,b^6\,e\right )\,\left (d+e\,x\right )+b^7\,d^2+a^2\,b^5\,e^2-2\,a\,b^6\,d\,e}+\frac {2\,B\,e\,{\left (d+e\,x\right )}^{5/2}}{5\,b^3}+\frac {7\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}\,\left (5\,A\,b\,e-9\,B\,a\,e+4\,B\,b\,d\right )}{-9\,B\,a^3\,e^4+22\,B\,a^2\,b\,d\,e^3+5\,A\,a^2\,b\,e^4-17\,B\,a\,b^2\,d^2\,e^2-10\,A\,a\,b^2\,d\,e^3+4\,B\,b^3\,d^3\,e+5\,A\,b^3\,d^2\,e^2}\right )\,{\left (a\,e-b\,d\right )}^{3/2}\,\left (5\,A\,b\,e-9\,B\,a\,e+4\,B\,b\,d\right )}{4\,b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x)

[Out]

((2*A*e^2 - 2*B*d*e)/(3*b^3) + (2*B*e*(3*b^3*d - 3*a*b^2*e))/(3*b^6))*(d + e*x)^(3/2) + ((((2*A*e^2 - 2*B*d*e)
/b^3 + (2*B*e*(3*b^3*d - 3*a*b^2*e))/b^6)*(3*b^3*d - 3*a*b^2*e))/b^3 - (6*B*e*(a*e - b*d)^2)/b^5)*(d + e*x)^(1
/2) - ((d + e*x)^(3/2)*(B*b^4*d^3*e - (17*B*a^3*b*e^4)/4 + (13*A*a^2*b^2*e^4)/4 + (13*A*b^4*d^2*e^2)/4 - (25*B
*a*b^3*d^2*e^2)/4 + (19*B*a^2*b^2*d*e^3)/2 - (13*A*a*b^3*d*e^3)/2) - (d + e*x)^(1/2)*((15*B*a^4*e^5)/4 - (11*A
*a^3*b*e^5)/4 + B*b^4*d^4*e + (11*A*b^4*d^3*e^2)/4 - (33*A*a*b^3*d^2*e^3)/4 + (33*A*a^2*b^2*d*e^4)/4 - (27*B*a
*b^3*d^3*e^2)/4 + (57*B*a^2*b^2*d^2*e^3)/4 - (49*B*a^3*b*d*e^4)/4))/(b^7*(d + e*x)^2 - (2*b^7*d - 2*a*b^6*e)*(
d + e*x) + b^7*d^2 + a^2*b^5*e^2 - 2*a*b^6*d*e) + (2*B*e*(d + e*x)^(5/2))/(5*b^3) + (7*e*atan((b^(1/2)*e*(a*e
- b*d)^(3/2)*(d + e*x)^(1/2)*(5*A*b*e - 9*B*a*e + 4*B*b*d))/(5*A*a^2*b*e^4 - 9*B*a^3*e^4 + 4*B*b^3*d^3*e + 5*A
*b^3*d^2*e^2 - 17*B*a*b^2*d^2*e^2 - 10*A*a*b^2*d*e^3 + 22*B*a^2*b*d*e^3))*(a*e - b*d)^(3/2)*(5*A*b*e - 9*B*a*e
 + 4*B*b*d))/(4*b^(11/2))

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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)**(7/2)/(b*x+a)**3,x)

[Out]

Timed out

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