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

Optimal. Leaf size=407 \[ -\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {7 e (a+b x) (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

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Rubi [A]  time = 0.38, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {770, 78, 47, 50, 63, 208} \begin {gather*} -\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (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 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (a+b x) (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} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d + e*x)^(3/2))/(12*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*
(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
((4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*
B)*(d + e*x)^(9/2))/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*
d + 5*A*b*e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(11/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^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]

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\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) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{8 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\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) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right ) (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{8 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\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) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^2 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{8 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\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) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^3 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{8 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\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) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 \left (b^2 d-a b e\right )^3 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\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) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 111, normalized size = 0.27 \begin {gather*} \frac {(a+b x) (d+e x)^{9/2} \left (\frac {e (a+b x)^2 (-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}+9 a B-9 A b\right )}{18 b \left ((a+b x)^2\right )^{3/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 64.93, size = 509, normalized size = 1.25 \begin {gather*} \frac {(-a e-b e x) \left (\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}}-\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}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(a*e) - b*e*x)*(-1/60*(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 + 50
4*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))/(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])))/(e*Sqrt[(a
*e + b*e*x)^2/e^2])

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fricas [A]  time = 0.47, size = 1060, normalized size = 2.60

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^2*x^2+2*a*b*x+a^2)^(3/2),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 = 0.44, size = 685, normalized size = 1.68 \begin {gather*} \frac {7 \, {\left (4 \, B b^{3} d^{3} e^{2} - 17 \, B a b^{2} d^{2} e^{3} + 5 \, A b^{3} d^{2} e^{3} + 22 \, B a^{2} b d e^{4} - 10 \, A a b^{2} d e^{4} - 9 \, B a^{3} e^{5} + 5 \, A a^{2} b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{4 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {{\left (4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{2} - 4 \, \sqrt {x e + d} B b^{4} d^{4} e^{2} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{3} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{3} + 27 \, \sqrt {x e + d} B a b^{3} d^{3} e^{3} - 11 \, \sqrt {x e + d} A b^{4} d^{3} e^{3} + 38 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{4} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{4} - 57 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{4} + 33 \, \sqrt {x e + d} A a b^{3} d^{2} e^{4} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{5} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{5} + 49 \, \sqrt {x e + d} B a^{3} b d e^{5} - 33 \, \sqrt {x e + d} A a^{2} b^{2} d e^{5} - 15 \, \sqrt {x e + d} B a^{4} e^{6} + 11 \, \sqrt {x e + d} A a^{3} b e^{6}\right )} e^{\left (-1\right )}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{12} e^{6} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{12} d e^{6} + 45 \, \sqrt {x e + d} B b^{12} d^{2} e^{6} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{11} e^{7} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{12} e^{7} - 135 \, \sqrt {x e + d} B a b^{11} d e^{7} + 45 \, \sqrt {x e + d} A b^{12} d e^{7} + 90 \, \sqrt {x e + d} B a^{2} b^{10} e^{8} - 45 \, \sqrt {x e + d} A a b^{11} e^{8}\right )} e^{\left (-5\right )}}{15 \, b^{15} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/4*(4*B*b^3*d^3*e^2 - 17*B*a*b^2*d^2*e^3 + 5*A*b^3*d^2*e^3 + 22*B*a^2*b*d*e^4 - 10*A*a*b^2*d*e^4 - 9*B*a^3*e^
5 + 5*A*a^2*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^(-1)/(sqrt(-b^2*d + a*b*e)*b^5*sgn((x*e + d)
*b*e - b*d*e + a*e^2)) - 1/4*(4*(x*e + d)^(3/2)*B*b^4*d^3*e^2 - 4*sqrt(x*e + d)*B*b^4*d^4*e^2 - 25*(x*e + d)^(
3/2)*B*a*b^3*d^2*e^3 + 13*(x*e + d)^(3/2)*A*b^4*d^2*e^3 + 27*sqrt(x*e + d)*B*a*b^3*d^3*e^3 - 11*sqrt(x*e + d)*
A*b^4*d^3*e^3 + 38*(x*e + d)^(3/2)*B*a^2*b^2*d*e^4 - 26*(x*e + d)^(3/2)*A*a*b^3*d*e^4 - 57*sqrt(x*e + d)*B*a^2
*b^2*d^2*e^4 + 33*sqrt(x*e + d)*A*a*b^3*d^2*e^4 - 17*(x*e + d)^(3/2)*B*a^3*b*e^5 + 13*(x*e + d)^(3/2)*A*a^2*b^
2*e^5 + 49*sqrt(x*e + d)*B*a^3*b*d*e^5 - 33*sqrt(x*e + d)*A*a^2*b^2*d*e^5 - 15*sqrt(x*e + d)*B*a^4*e^6 + 11*sq
rt(x*e + d)*A*a^3*b*e^6)*e^(-1)/(((x*e + d)*b - b*d + a*e)^2*b^5*sgn((x*e + d)*b*e - b*d*e + a*e^2)) + 2/15*(3
*(x*e + d)^(5/2)*B*b^12*e^6 + 10*(x*e + d)^(3/2)*B*b^12*d*e^6 + 45*sqrt(x*e + d)*B*b^12*d^2*e^6 - 15*(x*e + d)
^(3/2)*B*a*b^11*e^7 + 5*(x*e + d)^(3/2)*A*b^12*e^7 - 135*sqrt(x*e + d)*B*a*b^11*d*e^7 + 45*sqrt(x*e + d)*A*b^1
2*d*e^7 + 90*sqrt(x*e + d)*B*a^2*b^10*e^8 - 45*sqrt(x*e + d)*A*a*b^11*e^8)*e^(-5)/(b^15*sgn((x*e + d)*b*e - b*
d*e + a*e^2))

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maple [B]  time = 0.11, size = 1873, normalized size = 4.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(-155*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3+720*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a*b^3*d^2
*e^2+720*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a*b^3*d*e^3-2160*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^2*b^2*
d*e^3+160*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a*b^3*d*e^2-1080*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a*b^3
*d*e^3-945*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^5*e^5+60*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^4*d^
4+525*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^4*b*e^5-60*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^4*d^3+9
45*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*e^4+855*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3+360*A*((a
*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*b^4*d*e^3+80*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*b^4*d*e^2-1050*A*arcta
n((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a*b^4*d*e^4+48*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x*a*b^3*e^2-120*
B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*a*b^3*e^3-495*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2-1815*B
*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b*d*e^3+1215*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2-405*B*
((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e-3570*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^2*b^3*d^2
*e^3+840*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a*b^4*d^3*e^2+390*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)
*a*b^3*d*e^2-720*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^2*b^2*e^4-490*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2
*b^2*d*e^2+375*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e+1440*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*
b*e^4+720*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^2*e^4+360*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*b^4*
d^2*e^2+4620*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^3*b^2*d*e^4+1050*A*arctan((e*x+d)^(1/2)/((a*e-b
*d)*b)^(1/2)*b)*x*a*b^4*d^2*e^3-240*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a^2*b^2*e^3-2100*A*arctan((e*x+d)^(1
/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^2*b^3*d*e^4+2310*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^3*d*e^
4-1785*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a*b^4*d^2*e^3+80*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*
x*a*b^3*e^3-360*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a*b^3*e^4-525*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*
b*e^4+165*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^4*d^3*e+1050*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a
^3*b^2*e^5+24*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^2*b^2*e^2-1890*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*
b)*x*a^4*b*e^5+24*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x^2*b^4*e^2+40*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*b
^4*e^3+525*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^3*e^5+525*A*arctan((e*x+d)^(1/2)/((a*e-b*d)
*b)^(1/2)*b)*x^2*b^5*d^2*e^3-945*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^3*b^2*e^5+420*B*arctan((e
*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*b^5*d^3*e^2-1050*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^3*b^2*
d*e^4+525*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^2*b^3*d^2*e^3+135*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2
)*a^3*b*e^3+2310*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^4*b*d*e^4-1785*B*arctan((e*x+d)^(1/2)/((a*e-b
*d)*b)^(1/2)*b)*a^3*b^2*d^2*e^3+420*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^2*b^3*d^3*e^2-195*A*((a*e-
b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^4*d^2*e)/e*(b*x+a)/((a*e-b*d)*b)^(1/2)/b^5/((b*x+a)^2)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{7/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\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)^(7/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

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

[Out]

Timed out

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