3.17.61 \(\int \frac {A+B x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=424 \[ -\frac {35 e^3 (a+b x) (a B e-9 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}+\frac {35 e^3 (a+b x) (a B e-9 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}-\frac {35 e^2 (a B e-9 A b e+8 b B d)}{192 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}+\frac {7 e (a B e-9 A b e+8 b B d)}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-9 A b e+8 b B d}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}-\frac {A b-a B}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)} \]
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Rubi [A] time = 0.44, antiderivative size = 424, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used =
{770, 78, 51, 63, 208} \begin {gather*} -\frac {35 e^3 (a+b x) (a B e-9 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}-\frac {35 e^2 (a B e-9 A b e+8 b B d)}{192 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}+\frac {35 e^3 (a+b x) (a B e-9 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac {7 e (a B e-9 A b e+8 b B d)}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-9 A b e+8 b B d}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}-\frac {A b-a B}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-35*e^2*(8*b*B*d - 9*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (A*b
- a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 9*A*b*e + a*B*e
)/(24*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(8*b*B*d - 9*A*b*e + a*B
*e))/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d - 9*A*b*e +
a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d - 9*A*b
*e + a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d - a*e)^(11/2)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])
Rule 51
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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)^{3/2} \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}{\left (a b+b^2 x\right )^5 (d+e x)^{3/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d-9 A b e+a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 (d+e x)^{3/2}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b e (8 b B d-9 A b e+a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)^{3/2}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 (8 b B d-9 A b e+a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{3/2}} \, dx}{192 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 e^3 (8 b B d-9 A b e+a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{128 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 e^3 (8 b B d-9 A b e+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 d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 e^2 (8 b B d-9 A b e+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 d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 114, normalized size = 0.27 \begin {gather*} \frac {\frac {e^3 (a+b x)^4 (-a B e+9 A b e-8 b B d) \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+a B-A b}{4 b (a+b x)^3 \sqrt {(a+b x)^2} \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-(A*b) + a*B + (e^3*(-8*b*B*d + 9*A*b*e - a*B*e)*(a + b*x)^4*Hypergeometric2F1[-1/2, 4, 1/2, (b*(d + e*x))/(b
*d - a*e)])/(b*d - a*e)^4)/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2]*Sqrt[d + e*x])
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IntegrateAlgebraic [A] time = 63.85, size = 682, normalized size = 1.61 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {e^3 \left (384 a^4 A e^5-279 a^4 B e^4 (d+e x)-384 a^4 B d e^4+2511 a^3 A b e^4 (d+e x)-1536 a^3 A b d e^4+1536 a^3 b B d^2 e^3-1395 a^3 b B d e^3 (d+e x)-511 a^3 b B e^3 (d+e x)^2+2304 a^2 A b^2 d^2 e^3-7533 a^2 A b^2 d e^3 (d+e x)+4599 a^2 A b^2 e^3 (d+e x)^2-2304 a^2 b^2 B d^3 e^2+5859 a^2 b^2 B d^2 e^2 (d+e x)-3066 a^2 b^2 B d e^2 (d+e x)^2-385 a^2 b^2 B e^2 (d+e x)^3-1536 a A b^3 d^3 e^2+7533 a A b^3 d^2 e^2 (d+e x)-9198 a A b^3 d e^2 (d+e x)^2+3465 a A b^3 e^2 (d+e x)^3+1536 a b^3 B d^4 e-6417 a b^3 B d^3 e (d+e x)+7665 a b^3 B d^2 e (d+e x)^2-2695 a b^3 B d e (d+e x)^3-105 a b^3 B e (d+e x)^4+384 A b^4 d^4 e-2511 A b^4 d^3 e (d+e x)+4599 A b^4 d^2 e (d+e x)^2-3465 A b^4 d e (d+e x)^3+945 A b^4 e (d+e x)^4-384 b^4 B d^5+2232 b^4 B d^4 (d+e x)-4088 b^4 B d^3 (d+e x)^2+3080 b^4 B d^2 (d+e x)^3-840 b^4 B d (d+e x)^4\right )}{192 \sqrt {d+e x} (b d-a e)^5 (-a e-b (d+e x)+b d)^4}-\frac {35 \left (a B e^4-9 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 \sqrt {b} (b d-a e)^5 \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)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
((-(a*e) - b*e*x)*(-1/192*(e^3*(-384*b^4*B*d^5 + 384*A*b^4*d^4*e + 1536*a*b^3*B*d^4*e - 1536*a*A*b^3*d^3*e^2 -
2304*a^2*b^2*B*d^3*e^2 + 2304*a^2*A*b^2*d^2*e^3 + 1536*a^3*b*B*d^2*e^3 - 1536*a^3*A*b*d*e^4 - 384*a^4*B*d*e^4
+ 384*a^4*A*e^5 + 2232*b^4*B*d^4*(d + e*x) - 2511*A*b^4*d^3*e*(d + e*x) - 6417*a*b^3*B*d^3*e*(d + e*x) + 7533
*a*A*b^3*d^2*e^2*(d + e*x) + 5859*a^2*b^2*B*d^2*e^2*(d + e*x) - 7533*a^2*A*b^2*d*e^3*(d + e*x) - 1395*a^3*b*B*
d*e^3*(d + e*x) + 2511*a^3*A*b*e^4*(d + e*x) - 279*a^4*B*e^4*(d + e*x) - 4088*b^4*B*d^3*(d + e*x)^2 + 4599*A*b
^4*d^2*e*(d + e*x)^2 + 7665*a*b^3*B*d^2*e*(d + e*x)^2 - 9198*a*A*b^3*d*e^2*(d + e*x)^2 - 3066*a^2*b^2*B*d*e^2*
(d + e*x)^2 + 4599*a^2*A*b^2*e^3*(d + e*x)^2 - 511*a^3*b*B*e^3*(d + e*x)^2 + 3080*b^4*B*d^2*(d + e*x)^3 - 3465
*A*b^4*d*e*(d + e*x)^3 - 2695*a*b^3*B*d*e*(d + e*x)^3 + 3465*a*A*b^3*e^2*(d + e*x)^3 - 385*a^2*b^2*B*e^2*(d +
e*x)^3 - 840*b^4*B*d*(d + e*x)^4 + 945*A*b^4*e*(d + e*x)^4 - 105*a*b^3*B*e*(d + e*x)^4))/((b*d - a*e)^5*Sqrt[d
+ e*x]*(b*d - a*e - b*(d + e*x))^4) - (35*(8*b*B*d*e^3 - 9*A*b*e^4 + a*B*e^4)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a
*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*Sqrt[b]*(b*d - a*e)^5*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])
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fricas [B] time = 0.50, size = 2968, normalized size = 7.00
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(3/2)/(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^2*e^3 + (B*a^5 - 9*A*a^4*b)*d*e^4 + (8*B*b^5*d*e^4 + (B*a*b^4 - 9*A*b^5)*e^5)*x^5 + (
8*B*b^5*d^2*e^3 + 3*(11*B*a*b^4 - 3*A*b^5)*d*e^4 + 4*(B*a^2*b^3 - 9*A*a*b^4)*e^5)*x^4 + 2*(16*B*a*b^4*d^2*e^3
+ 2*(13*B*a^2*b^3 - 9*A*a*b^4)*d*e^4 + 3*(B*a^3*b^2 - 9*A*a^2*b^3)*e^5)*x^3 + 2*(24*B*a^2*b^3*d^2*e^3 + (19*B*
a^3*b^2 - 27*A*a^2*b^3)*d*e^4 + 2*(B*a^4*b - 9*A*a^3*b^2)*e^5)*x^2 + (32*B*a^3*b^2*d^2*e^3 + 12*(B*a^4*b - 3*A
*a^3*b^2)*d*e^4 + (B*a^5 - 9*A*a^4*b)*e^5)*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*(384*A*a^5*b*e^5 + 16*(B*a*b^5 + 3*A*b^6)*d^5 - 24*(5*B*a^2*b^4 + 13*A*a*b^
5)*d^4*e + 6*(79*B*a^3*b^3 + 149*A*a^2*b^4)*d^3*e^2 + (293*B*a^4*b^2 - 1605*A*a^3*b^3)*d^2*e^3 - 3*(221*B*a^5*
b - 197*A*a^4*b^2)*d*e^4 + 105*(8*B*b^6*d^2*e^3 - (7*B*a*b^5 + 9*A*b^6)*d*e^4 - (B*a^2*b^4 - 9*A*a*b^5)*e^5)*x
^4 + 35*(8*B*b^6*d^3*e^2 + 9*(9*B*a*b^5 - A*b^6)*d^2*e^3 - 6*(13*B*a^2*b^4 + 15*A*a*b^5)*d*e^4 - 11*(B*a^3*b^3
- 9*A*a^2*b^4)*e^5)*x^3 - 7*(16*B*b^6*d^4*e - 2*(83*B*a*b^5 + 9*A*b^6)*d^3*e^2 - 3*(151*B*a^2*b^4 - 63*A*a*b^
5)*d^2*e^3 + 2*(265*B*a^3*b^3 + 243*A*a^2*b^4)*d*e^4 + 73*(B*a^4*b^2 - 9*A*a^3*b^3)*e^5)*x^2 + (64*B*b^6*d^5 -
8*(59*B*a*b^5 + 9*A*b^6)*d^4*e + 108*(17*B*a^2*b^4 + 5*A*a*b^5)*d^3*e^2 + (989*B*a^3*b^3 - 2133*A*a^2*b^4)*d^
2*e^3 - 2*(1069*B*a^4*b^2 + 423*A*a^3*b^3)*d*e^4 - 279*(B*a^5*b - 9*A*a^4*b^2)*e^5)*x)*sqrt(e*x + d))/(a^4*b^7
*d^7 - 6*a^5*b^6*d^6*e + 15*a^6*b^5*d^5*e^2 - 20*a^7*b^4*d^4*e^3 + 15*a^8*b^3*d^3*e^4 - 6*a^9*b^2*d^2*e^5 + a^
10*b*d*e^6 + (b^11*d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6
*a^5*b^6*d*e^6 + a^6*b^5*e^7)*x^5 + (b^11*d^7 - 2*a*b^10*d^6*e - 9*a^2*b^9*d^5*e^2 + 40*a^3*b^8*d^4*e^3 - 65*a
^4*b^7*d^3*e^4 + 54*a^5*b^6*d^2*e^5 - 23*a^6*b^5*d*e^6 + 4*a^7*b^4*e^7)*x^4 + 2*(2*a*b^10*d^7 - 9*a^2*b^9*d^6*
e + 12*a^3*b^8*d^5*e^2 + 5*a^4*b^7*d^4*e^3 - 30*a^5*b^6*d^3*e^4 + 33*a^6*b^5*d^2*e^5 - 16*a^7*b^4*d*e^6 + 3*a^
8*b^3*e^7)*x^3 + 2*(3*a^2*b^9*d^7 - 16*a^3*b^8*d^6*e + 33*a^4*b^7*d^5*e^2 - 30*a^5*b^6*d^4*e^3 + 5*a^6*b^5*d^3
*e^4 + 12*a^7*b^4*d^2*e^5 - 9*a^8*b^3*d*e^6 + 2*a^9*b^2*e^7)*x^2 + (4*a^3*b^8*d^7 - 23*a^4*b^7*d^6*e + 54*a^5*
b^6*d^5*e^2 - 65*a^6*b^5*d^4*e^3 + 40*a^7*b^4*d^3*e^4 - 9*a^8*b^3*d^2*e^5 - 2*a^9*b^2*d*e^6 + a^10*b*e^7)*x),
-1/192*(105*(8*B*a^4*b*d^2*e^3 + (B*a^5 - 9*A*a^4*b)*d*e^4 + (8*B*b^5*d*e^4 + (B*a*b^4 - 9*A*b^5)*e^5)*x^5 + (
8*B*b^5*d^2*e^3 + 3*(11*B*a*b^4 - 3*A*b^5)*d*e^4 + 4*(B*a^2*b^3 - 9*A*a*b^4)*e^5)*x^4 + 2*(16*B*a*b^4*d^2*e^3
+ 2*(13*B*a^2*b^3 - 9*A*a*b^4)*d*e^4 + 3*(B*a^3*b^2 - 9*A*a^2*b^3)*e^5)*x^3 + 2*(24*B*a^2*b^3*d^2*e^3 + (19*B*
a^3*b^2 - 27*A*a^2*b^3)*d*e^4 + 2*(B*a^4*b - 9*A*a^3*b^2)*e^5)*x^2 + (32*B*a^3*b^2*d^2*e^3 + 12*(B*a^4*b - 3*A
*a^3*b^2)*d*e^4 + (B*a^5 - 9*A*a^4*b)*e^5)*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)) + (384*A*a^5*b*e^5 + 16*(B*a*b^5 + 3*A*b^6)*d^5 - 24*(5*B*a^2*b^4 + 13*A*a*b^5)*d^4*e + 6*(79*B*
a^3*b^3 + 149*A*a^2*b^4)*d^3*e^2 + (293*B*a^4*b^2 - 1605*A*a^3*b^3)*d^2*e^3 - 3*(221*B*a^5*b - 197*A*a^4*b^2)*
d*e^4 + 105*(8*B*b^6*d^2*e^3 - (7*B*a*b^5 + 9*A*b^6)*d*e^4 - (B*a^2*b^4 - 9*A*a*b^5)*e^5)*x^4 + 35*(8*B*b^6*d^
3*e^2 + 9*(9*B*a*b^5 - A*b^6)*d^2*e^3 - 6*(13*B*a^2*b^4 + 15*A*a*b^5)*d*e^4 - 11*(B*a^3*b^3 - 9*A*a^2*b^4)*e^5
)*x^3 - 7*(16*B*b^6*d^4*e - 2*(83*B*a*b^5 + 9*A*b^6)*d^3*e^2 - 3*(151*B*a^2*b^4 - 63*A*a*b^5)*d^2*e^3 + 2*(265
*B*a^3*b^3 + 243*A*a^2*b^4)*d*e^4 + 73*(B*a^4*b^2 - 9*A*a^3*b^3)*e^5)*x^2 + (64*B*b^6*d^5 - 8*(59*B*a*b^5 + 9*
A*b^6)*d^4*e + 108*(17*B*a^2*b^4 + 5*A*a*b^5)*d^3*e^2 + (989*B*a^3*b^3 - 2133*A*a^2*b^4)*d^2*e^3 - 2*(1069*B*a
^4*b^2 + 423*A*a^3*b^3)*d*e^4 - 279*(B*a^5*b - 9*A*a^4*b^2)*e^5)*x)*sqrt(e*x + d))/(a^4*b^7*d^7 - 6*a^5*b^6*d^
6*e + 15*a^6*b^5*d^5*e^2 - 20*a^7*b^4*d^4*e^3 + 15*a^8*b^3*d^3*e^4 - 6*a^9*b^2*d^2*e^5 + a^10*b*d*e^6 + (b^11*
d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6*a^5*b^6*d*e^6 + a^
6*b^5*e^7)*x^5 + (b^11*d^7 - 2*a*b^10*d^6*e - 9*a^2*b^9*d^5*e^2 + 40*a^3*b^8*d^4*e^3 - 65*a^4*b^7*d^3*e^4 + 54
*a^5*b^6*d^2*e^5 - 23*a^6*b^5*d*e^6 + 4*a^7*b^4*e^7)*x^4 + 2*(2*a*b^10*d^7 - 9*a^2*b^9*d^6*e + 12*a^3*b^8*d^5*
e^2 + 5*a^4*b^7*d^4*e^3 - 30*a^5*b^6*d^3*e^4 + 33*a^6*b^5*d^2*e^5 - 16*a^7*b^4*d*e^6 + 3*a^8*b^3*e^7)*x^3 + 2*
(3*a^2*b^9*d^7 - 16*a^3*b^8*d^6*e + 33*a^4*b^7*d^5*e^2 - 30*a^5*b^6*d^4*e^3 + 5*a^6*b^5*d^3*e^4 + 12*a^7*b^4*d
^2*e^5 - 9*a^8*b^3*d*e^6 + 2*a^9*b^2*e^7)*x^2 + (4*a^3*b^8*d^7 - 23*a^4*b^7*d^6*e + 54*a^5*b^6*d^5*e^2 - 65*a^
6*b^5*d^4*e^3 + 40*a^7*b^4*d^3*e^4 - 9*a^8*b^3*d^2*e^5 - 2*a^9*b^2*d*e^6 + a^10*b*e^7)*x)]
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giac [B] time = 0.63, size = 1132, normalized size = 2.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
-35/64*(8*B*b*d*e^3 + B*a*e^4 - 9*A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5*sgn((x*e + d
)*b*e - b*d*e + a*e^2) - 5*a*b^4*d^4*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 10*a^2*b^3*d^3*e^2*sgn((x*e + d)*b
*e - b*d*e + a*e^2) - 10*a^3*b^2*d^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 5*a^4*b*d*e^4*sgn((x*e + d)*b*e
- b*d*e + a*e^2) - a^5*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 2*(B*d*e^3 - A*e^4)/((b
^5*d^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 5*a*b^4*d^4*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 10*a^2*b^3*d^3*
e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 10*a^3*b^2*d^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 5*a^4*b*d*e^4
*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^5*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(x*e + d)) - 1/192*(456*
(x*e + d)^(7/2)*B*b^4*d*e^3 - 1544*(x*e + d)^(5/2)*B*b^4*d^2*e^3 + 1784*(x*e + d)^(3/2)*B*b^4*d^3*e^3 - 696*sq
rt(x*e + d)*B*b^4*d^4*e^3 + 105*(x*e + d)^(7/2)*B*a*b^3*e^4 - 561*(x*e + d)^(7/2)*A*b^4*e^4 + 1159*(x*e + d)^(
5/2)*B*a*b^3*d*e^4 + 1929*(x*e + d)^(5/2)*A*b^4*d*e^4 - 3057*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 - 2295*(x*e + d)^
(3/2)*A*b^4*d^2*e^4 + 1809*sqrt(x*e + d)*B*a*b^3*d^3*e^4 + 975*sqrt(x*e + d)*A*b^4*d^3*e^4 + 385*(x*e + d)^(5/
2)*B*a^2*b^2*e^5 - 1929*(x*e + d)^(5/2)*A*a*b^3*e^5 + 762*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 + 4590*(x*e + d)^(3/
2)*A*a*b^3*d*e^5 - 1251*sqrt(x*e + d)*B*a^2*b^2*d^2*e^5 - 2925*sqrt(x*e + d)*A*a*b^3*d^2*e^5 + 511*(x*e + d)^(
3/2)*B*a^3*b*e^6 - 2295*(x*e + d)^(3/2)*A*a^2*b^2*e^6 - 141*sqrt(x*e + d)*B*a^3*b*d*e^6 + 2925*sqrt(x*e + d)*A
*a^2*b^2*d*e^6 + 279*sqrt(x*e + d)*B*a^4*e^7 - 975*sqrt(x*e + d)*A*a^3*b*e^7)/((b^5*d^5*sgn((x*e + d)*b*e - b*
d*e + a*e^2) - 5*a*b^4*d^4*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 10*a^2*b^3*d^3*e^2*sgn((x*e + d)*b*e - b*d*e
+ a*e^2) - 10*a^3*b^2*d^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 5*a^4*b*d*e^4*sgn((x*e + d)*b*e - b*d*e +
a*e^2) - a^5*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)
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maple [B] time = 0.12, size = 1493, normalized size = 3.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
-1/192*(945*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*a^4*b*e^4-48*((a*e-b*d)*b)^(1/2)*A*b^4
*d^4-663*((a*e-b*d)*b)^(1/2)*B*a^4*d*e^3-3360*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^3*
a*b^4*d*e^3-5040*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^2*a^2*b^3*d*e^3-3360*B*arctan((
e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x*a^3*b^2*d*e^3+384*((a*e-b*d)*b)^(1/2)*A*a^4*e^4+945*((a*e-
b*d)*b)^(1/2)*A*b^4*e^4*x^4-279*((a*e-b*d)*b)^(1/2)*B*a^4*e^4*x-64*((a*e-b*d)*b)^(1/2)*B*b^4*d^4*x-16*((a*e-b*
d)*b)^(1/2)*B*a*b^3*d^4-105*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*a^5*e^4-840*B*arctan((
e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^4*b^5*d*e^3+3780*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2
)*b)*(e*x+d)^(1/2)*x^3*a*b^4*e^4-420*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^3*a^2*b^3*e
^4+5670*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^2*a^2*b^3*e^4-630*B*arctan((e*x+d)^(1/2)
/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^2*a^3*b^2*e^4+3780*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+
d)^(1/2)*x*a^3*b^2*e^4-420*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x*a^4*b*e^4-840*B*arcta
n((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*a^4*b*d*e^3-105*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2
)*b)*(e*x+d)^(1/2)*x^4*a*b^4*e^4+945*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^4*b^5*e^4+3
465*((a*e-b*d)*b)^(1/2)*A*a*b^3*e^4*x^3+315*((a*e-b*d)*b)^(1/2)*A*b^4*d*e^3*x^3-385*((a*e-b*d)*b)^(1/2)*B*a^2*
b^2*e^4*x^3-280*((a*e-b*d)*b)^(1/2)*B*b^4*d^2*e^2*x^3+4599*((a*e-b*d)*b)^(1/2)*A*a^2*b^2*e^4*x^2-126*((a*e-b*d
)*b)^(1/2)*A*b^4*d^2*e^2*x^2-511*((a*e-b*d)*b)^(1/2)*B*a^3*b*e^4*x^2+112*((a*e-b*d)*b)^(1/2)*B*b^4*d^3*e*x^2+2
511*((a*e-b*d)*b)^(1/2)*A*a^3*b*e^4*x+72*((a*e-b*d)*b)^(1/2)*A*b^4*d^3*e*x-105*((a*e-b*d)*b)^(1/2)*B*a*b^3*e^4
*x^4-840*((a*e-b*d)*b)^(1/2)*B*b^4*d*e^3*x^4+975*((a*e-b*d)*b)^(1/2)*A*a^3*b*d*e^3-630*((a*e-b*d)*b)^(1/2)*A*a
^2*b^2*d^2*e^2+264*((a*e-b*d)*b)^(1/2)*A*a*b^3*d^3*e-370*((a*e-b*d)*b)^(1/2)*B*a^3*b*d^2*e^2+104*((a*e-b*d)*b)
^(1/2)*B*a^2*b^2*d^3*e-4221*((a*e-b*d)*b)^(1/2)*B*a^2*b^2*d*e^3*x^2-1050*((a*e-b*d)*b)^(1/2)*B*a*b^3*d^2*e^2*x
^2+1665*((a*e-b*d)*b)^(1/2)*A*a^2*b^2*d*e^3*x-468*((a*e-b*d)*b)^(1/2)*A*a*b^3*d^2*e^2*x-2417*((a*e-b*d)*b)^(1/
2)*B*a^3*b*d*e^3*x-1428*((a*e-b*d)*b)^(1/2)*B*a^2*b^2*d^2*e^2*x+408*((a*e-b*d)*b)^(1/2)*B*a*b^3*d^3*e*x-3115*(
(a*e-b*d)*b)^(1/2)*B*a*b^3*d*e^3*x^3+1197*((a*e-b*d)*b)^(1/2)*A*a*b^3*d*e^3*x^2)*(b*x+a)/(e*x+d)^(1/2)/((a*e-b
*d)*b)^(1/2)/(a*e-b*d)^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 {B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\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)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(3/2)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^{3/2}\,{\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)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
[Out]
int((A + B*x)/((d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
________________________________________________________________________________________
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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Timed out
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