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

Optimal. Leaf size=424 \[ -\frac {(d+e x)^{9/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {35 e^3 (a+b x) (-9 a B e+A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {b d-a e}}+\frac {35 e^3 (a+b x) \sqrt {d+e x} (-9 a B e+A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {35 e^2 (d+e x)^{3/2} (-9 a B e+A b e+8 b B d)}{192 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e+A b e+8 b B d)}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

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Rubi [A]  time = 0.35, antiderivative size = 424, 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 {35 e^2 (d+e x)^{3/2} (-9 a B e+A b e+8 b B d)}{192 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {35 e^3 (a+b x) \sqrt {d+e x} (-9 a B e+A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {35 e^3 (a+b x) (-9 a B e+A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {b d-a e}}-\frac {(d+e x)^{9/2} (A b-a B)}{4 b (a+b x)^3 \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+A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e+A b e+8 b B d)}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \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)^(5/2),x]

[Out]

(35*e^3*(8*b*B*d + A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - (35*e^2*(8*b*B*d + A*b*e - 9*a*B*e)*(d + e*x)^(3/2))/(192*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(7*e*(8*b*B*d + A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(96*b^3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - ((8*b*B*d + A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - ((A*b - a*B)*(d + e*x)^(9/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*
B*d + A*b*e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(11/2)*Sqrt[b*d - a*e
]*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 )^{5/2}} \, dx &=\frac {\left (b^4 \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 )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d+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 )^4} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e (8 b B d+A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 (8 b B d+A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{192 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^3 (8 b B d+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}{128 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+A b e-9 a B e) (a+b x) \sqrt {d+e x}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^3 \left (b^2 d-a b e\right ) (8 b B d+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}{128 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+A b e-9 a B e) (a+b x) \sqrt {d+e x}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (b^2 d-a b e\right ) (8 b B d+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 )}{64 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+A b e-9 a B e) (a+b x) \sqrt {d+e x}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d+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 )}{64 b^{11/2} \sqrt {b d-a e} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 114, normalized size = 0.27 \begin {gather*} \frac {(d+e x)^{9/2} \left (\frac {e^3 (a+b x)^4 (-9 a B e+A b e+8 b B d) \, _2F_1\left (4,\frac {9}{2};\frac {11}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+9 a B-9 A b\right )}{36 b (a+b x)^3 \sqrt {(a+b x)^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)^(5/2),x]

[Out]

((d + e*x)^(9/2)*(-9*A*b + 9*a*B + (e^3*(8*b*B*d + A*b*e - 9*a*B*e)*(a + b*x)^4*Hypergeometric2F1[4, 9/2, 11/2
, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(36*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 64.73, size = 502, normalized size = 1.18 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e^3 \sqrt {d+e x} \left (-945 a^4 B e^4+105 a^3 A b e^4-3465 a^3 b B e^3 (d+e x)+3675 a^3 b B d e^3+385 a^2 A b^2 e^3 (d+e x)-315 a^2 A b^2 d e^3-5355 a^2 b^2 B d^2 e^2-4599 a^2 b^2 B e^2 (d+e x)^2+10010 a^2 b^2 B d e^2 (d+e x)+315 a A b^3 d^2 e^2+511 a A b^3 e^2 (d+e x)^2-770 a A b^3 d e^2 (d+e x)+3465 a b^3 B d^3 e-9625 a b^3 B d^2 e (d+e x)-2511 a b^3 B e (d+e x)^3+8687 a b^3 B d e (d+e x)^2-105 A b^4 d^3 e+385 A b^4 d^2 e (d+e x)+279 A b^4 e (d+e x)^3-511 A b^4 d e (d+e x)^2-840 b^4 B d^4+3080 b^4 B d^3 (d+e x)-4088 b^4 B d^2 (d+e x)^2-384 b^4 B (d+e x)^4+2232 b^4 B d (d+e x)^3\right )}{192 b^5 (a e+b (d+e x)-b d)^4}+\frac {35 \left (-9 a B e^4+A b e^4+8 b B d e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{11/2} \sqrt {a e-b d}}\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)^(5/2),x]

[Out]

((-(a*e) - b*e*x)*((e^3*Sqrt[d + e*x]*(-840*b^4*B*d^4 - 105*A*b^4*d^3*e + 3465*a*b^3*B*d^3*e + 315*a*A*b^3*d^2
*e^2 - 5355*a^2*b^2*B*d^2*e^2 - 315*a^2*A*b^2*d*e^3 + 3675*a^3*b*B*d*e^3 + 105*a^3*A*b*e^4 - 945*a^4*B*e^4 + 3
080*b^4*B*d^3*(d + e*x) + 385*A*b^4*d^2*e*(d + e*x) - 9625*a*b^3*B*d^2*e*(d + e*x) - 770*a*A*b^3*d*e^2*(d + e*
x) + 10010*a^2*b^2*B*d*e^2*(d + e*x) + 385*a^2*A*b^2*e^3*(d + e*x) - 3465*a^3*b*B*e^3*(d + e*x) - 4088*b^4*B*d
^2*(d + e*x)^2 - 511*A*b^4*d*e*(d + e*x)^2 + 8687*a*b^3*B*d*e*(d + e*x)^2 + 511*a*A*b^3*e^2*(d + e*x)^2 - 4599
*a^2*b^2*B*e^2*(d + e*x)^2 + 2232*b^4*B*d*(d + e*x)^3 + 279*A*b^4*e*(d + e*x)^3 - 2511*a*b^3*B*e*(d + e*x)^3 -
384*b^4*B*(d + e*x)^4))/(192*b^5*(-(b*d) + a*e + b*(d + e*x))^4) + (35*(8*b*B*d*e^3 + A*b*e^4 - 9*a*B*e^4)*Ar
cTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*b^(11/2)*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e
+ b*e*x)^2/e^2])

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fricas [B]  time = 0.47, size = 1485, normalized size = 3.50

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)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(105*(8*B*a^4*b*d*e^3 - (9*B*a^5 - A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (9*B*a*b^4 - A*b^5)*e^4)*x^4 + 4*(8*
B*a*b^4*d*e^3 - (9*B*a^2*b^3 - A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (9*B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 +
4*(8*B*a^3*b^2*d*e^3 - (9*B*a^4*b - A*a^3*b^2)*e^4)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(
b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(16*(B*a*b^5 + 3*A*b^6)*d^4 + 8*(5*B*a^2*b^4 + A*a*b^5)*d^3*e + 1
4*(11*B*a^3*b^3 + A*a^2*b^4)*d^2*e^2 - 35*(33*B*a^4*b^2 - A*a^3*b^3)*d*e^3 + 105*(9*B*a^5*b - A*a^4*b^2)*e^4 -
384*(B*b^6*d*e^3 - B*a*b^5*e^4)*x^4 + 3*(232*B*b^6*d^2*e^2 - (1069*B*a*b^5 - 93*A*b^6)*d*e^3 + 93*(9*B*a^2*b^
4 - A*a*b^5)*e^4)*x^3 + (304*B*b^6*d^3*e + 2*(425*B*a*b^5 + 163*A*b^6)*d^2*e^2 - (5753*B*a^2*b^4 - 185*A*a*b^5
)*d*e^3 + 511*(9*B*a^3*b^3 - A*a^2*b^4)*e^4)*x^2 + (64*B*b^6*d^4 + 8*(19*B*a*b^5 + 25*A*b^6)*d^3*e + 4*(149*B*
a^2*b^4 + 13*A*a*b^5)*d^2*e^2 - 7*(611*B*a^3*b^3 - 19*A*a^2*b^4)*d*e^3 + 385*(9*B*a^4*b^2 - A*a^3*b^3)*e^4)*x)
*sqrt(e*x + d))/(a^4*b^7*d - a^5*b^6*e + (b^11*d - a*b^10*e)*x^4 + 4*(a*b^10*d - a^2*b^9*e)*x^3 + 6*(a^2*b^9*d
- a^3*b^8*e)*x^2 + 4*(a^3*b^8*d - a^4*b^7*e)*x), 1/192*(105*(8*B*a^4*b*d*e^3 - (9*B*a^5 - A*a^4*b)*e^4 + (8*B
*b^5*d*e^3 - (9*B*a*b^4 - A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (9*B*a^2*b^3 - A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2
*b^3*d*e^3 - (9*B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (9*B*a^4*b - A*a^3*b^2)*e^4)*x)*sqrt(
-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (16*(B*a*b^5 + 3*A*b^6)*d^4 + 8*(5*
B*a^2*b^4 + A*a*b^5)*d^3*e + 14*(11*B*a^3*b^3 + A*a^2*b^4)*d^2*e^2 - 35*(33*B*a^4*b^2 - A*a^3*b^3)*d*e^3 + 105
*(9*B*a^5*b - A*a^4*b^2)*e^4 - 384*(B*b^6*d*e^3 - B*a*b^5*e^4)*x^4 + 3*(232*B*b^6*d^2*e^2 - (1069*B*a*b^5 - 93
*A*b^6)*d*e^3 + 93*(9*B*a^2*b^4 - A*a*b^5)*e^4)*x^3 + (304*B*b^6*d^3*e + 2*(425*B*a*b^5 + 163*A*b^6)*d^2*e^2 -
(5753*B*a^2*b^4 - 185*A*a*b^5)*d*e^3 + 511*(9*B*a^3*b^3 - A*a^2*b^4)*e^4)*x^2 + (64*B*b^6*d^4 + 8*(19*B*a*b^5
+ 25*A*b^6)*d^3*e + 4*(149*B*a^2*b^4 + 13*A*a*b^5)*d^2*e^2 - 7*(611*B*a^3*b^3 - 19*A*a^2*b^4)*d*e^3 + 385*(9*
B*a^4*b^2 - A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d - a^5*b^6*e + (b^11*d - a*b^10*e)*x^4 + 4*(a*b^10*d -
a^2*b^9*e)*x^3 + 6*(a^2*b^9*d - a^3*b^8*e)*x^2 + 4*(a^3*b^8*d - a^4*b^7*e)*x)]

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giac [A]  time = 0.49, size = 620, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{3}}{b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {35 \, {\left (8 \, B b d e^{3} - 9 \, B a e^{4} + A b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \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 {696 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} d e^{3} - 1784 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} + 1544 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{3} - 456 \, \sqrt {x e + d} B b^{4} d^{4} e^{3} - 975 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} + 279 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{4} + 4079 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} - 511 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} - 5017 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} + 385 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} + 1929 \, \sqrt {x e + d} B a b^{3} d^{3} e^{4} - 105 \, \sqrt {x e + d} A b^{4} d^{3} e^{4} - 2295 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} + 511 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} + 5402 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} - 770 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} - 3051 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{5} + 315 \, \sqrt {x e + d} A a b^{3} d^{2} e^{5} - 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} + 385 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} + 2139 \, \sqrt {x e + d} B a^{3} b d e^{6} - 315 \, \sqrt {x e + d} A a^{2} b^{2} d e^{6} - 561 \, \sqrt {x e + d} B a^{4} e^{7} + 105 \, \sqrt {x e + d} A a^{3} b e^{7}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5} \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)^(5/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*e^3/(b^5*sgn((x*e + d)*b*e - b*d*e + a*e^2)) + 35/64*(8*B*b*d*e^3 - 9*B*a*e^4 + A*b*e^4)*arc
tan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5*sgn((x*e + d)*b*e - b*d*e + a*e^2)) - 1/19
2*(696*(x*e + d)^(7/2)*B*b^4*d*e^3 - 1784*(x*e + d)^(5/2)*B*b^4*d^2*e^3 + 1544*(x*e + d)^(3/2)*B*b^4*d^3*e^3 -
456*sqrt(x*e + d)*B*b^4*d^4*e^3 - 975*(x*e + d)^(7/2)*B*a*b^3*e^4 + 279*(x*e + d)^(7/2)*A*b^4*e^4 + 4079*(x*e
+ d)^(5/2)*B*a*b^3*d*e^4 - 511*(x*e + d)^(5/2)*A*b^4*d*e^4 - 5017*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 + 385*(x*e
+ d)^(3/2)*A*b^4*d^2*e^4 + 1929*sqrt(x*e + d)*B*a*b^3*d^3*e^4 - 105*sqrt(x*e + d)*A*b^4*d^3*e^4 - 2295*(x*e +
d)^(5/2)*B*a^2*b^2*e^5 + 511*(x*e + d)^(5/2)*A*a*b^3*e^5 + 5402*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 - 770*(x*e + d
)^(3/2)*A*a*b^3*d*e^5 - 3051*sqrt(x*e + d)*B*a^2*b^2*d^2*e^5 + 315*sqrt(x*e + d)*A*a*b^3*d^2*e^5 - 1929*(x*e +
d)^(3/2)*B*a^3*b*e^6 + 385*(x*e + d)^(3/2)*A*a^2*b^2*e^6 + 2139*sqrt(x*e + d)*B*a^3*b*d*e^6 - 315*sqrt(x*e +
d)*A*a^2*b^2*d*e^6 - 561*sqrt(x*e + d)*B*a^4*e^7 + 105*sqrt(x*e + d)*A*a^3*b*e^7)/(((x*e + d)*b - b*d + a*e)^4
*b^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))

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maple [B]  time = 0.08, size = 1390, normalized size = 3.28

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)^(5/2),x)

[Out]

-1/192*(385*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*A*a^2*b^2*e^3+945*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*
x^4*a*b^4*e^5+945*B*a^5*e^5*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-456*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*
B*b^4*d^4-105*A*a^4*b*e^5*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+1544*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*B
*b^4*d^3-105*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*b^5*e^5+279*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)
*b^4*e+696*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^4*d-1784*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^4*d^2-945*((a*
e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a^4*e^4-315*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*A*a^2*b^2*d*e^3+315*((a*e-b*d)*b
)^(1/2)*(e*x+d)^(1/2)*A*a*b^3*d^2*e^2+2139*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a^3*b*d*e^3-3051*((a*e-b*d)*b)^
(1/2)*(e*x+d)^(1/2)*B*a^2*b^2*d^2*e^2+1929*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a*b^3*d^3*e-770*((a*e-b*d)*b)^(
1/2)*(e*x+d)^(3/2)*A*a*b^3*d*e^2+5402*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*B*a^2*b^2*d*e^2-5017*((a*e-b*d)*b)^(1/
2)*(e*x+d)^(3/2)*B*a*b^3*d^2*e-1536*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a^3*b*e^4*x-2304*((a*e-b*d)*b)^(1/2)*(
e*x+d)^(1/2)*B*a^2*b^2*e^4*x^2-3360*B*a^3*b^2*d*e^4*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-5040*B*a^2*b
^3*d*e^4*x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+4079*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^3*d*e-33
60*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^4*d*e^4-1536*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*
a*b^3*e^4+105*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*A*a^3*b*e^4-105*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*A*b^4*d^3*e-
420*A*a^3*b^2*e^5*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-2295*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*B*a^2*b
^2*e^2+3780*B*a^4*b*e^5*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-630*A*a^2*b^3*e^5*x^2*arctan((e*x+d)^(1/
2)/((a*e-b*d)*b)^(1/2)*b)+5670*B*a^3*b^2*e^5*x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-1929*((a*e-b*d)*b
)^(1/2)*(e*x+d)^(3/2)*B*a^3*b*e^3-840*B*a^4*b*d*e^4*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+385*((a*e-b*d)
*b)^(1/2)*(e*x+d)^(3/2)*A*b^4*d^2*e+511*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2-511*A*((a*e-b*d)*b)^(1/2
)*(e*x+d)^(5/2)*b^4*d*e-840*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*b^5*d*e^4-420*A*arctan((e*x+d)^(
1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^4*e^5-384*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*b^4*e^4+3780*B*arctan((e
*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a^2*b^3*e^5-975*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^3*e)/e*(b*x+a)/
((a*e-b*d)*b)^(1/2)/b^5/((b*x+a)^2)^(5/2)

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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 {5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), 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 )}^{5/2}} \,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)^(5/2),x)

[Out]

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

[Out]

Timed out

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