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

Optimal. Leaf size=164 \[ -\frac {2 c^{3/2} (b B-A c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{5/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{d^2 \sqrt {d+e x} (c d-b e)^2}+\frac {2 (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \]

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Rubi [A]  time = 0.32, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {828, 826, 1166, 208} \begin {gather*} -\frac {2 c^{3/2} (b B-A c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{5/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{d^2 \sqrt {d+e x} (c d-b e)^2}+\frac {2 (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*d - A*e))/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) + (2*(B*c*d^2 - A*e*(2*c*d - b*e)))/(d^2*(c*d - b*e)^2*Sqrt[
d + e*x]) - (2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(5/2)) - (2*c^(3/2)*(b*B - A*c)*ArcTanh[(Sqrt[c]*Sqrt[d
+ e*x])/Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(5/2))

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 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

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

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Mathematica [C]  time = 0.03, size = 91, normalized size = 0.55 \begin {gather*} \frac {2 \left (d (b B-A c) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )+A (c d-b e) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e x}{d}+1\right )\right )}{3 b d (d+e x)^{3/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((b*B - A*c)*d*Hypergeometric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] + A*(c*d - b*e)*Hypergeometric2F
1[-3/2, 1, -1/2, 1 + (e*x)/d]))/(3*b*d*(c*d - b*e)*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.48, size = 191, normalized size = 1.16 \begin {gather*} \frac {2 \left (A c^{5/2}-b B c^{3/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b (b e-c d)^{5/2}}+\frac {2 \left (3 A b e^2 (d+e x)+A b d e^2-A c d^2 e-6 A c d e (d+e x)-b B d^2 e+B c d^3+3 B c d^2 (d+e x)\right )}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*c*d^3 - b*B*d^2*e - A*c*d^2*e + A*b*d*e^2 + 3*B*c*d^2*(d + e*x) - 6*A*c*d*e*(d + e*x) + 3*A*b*e^2*(d + e
*x)))/(3*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) + (2*(-(b*B*c^(3/2)) + A*c^(5/2))*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*
e]*Sqrt[d + e*x])/(c*d - b*e)])/(b*(-(c*d) + b*e)^(5/2)) - (2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(5/2))

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fricas [B]  time = 2.06, size = 1709, normalized size = 10.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

[-1/3*(3*((B*b*c - A*c^2)*d^3*e^2*x^2 + 2*(B*b*c - A*c^2)*d^4*e*x + (B*b*c - A*c^2)*d^5)*sqrt(c/(c*d - b*e))*l
og((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 3*(A*c^2*d^4 - 2*A*b*c
*d^3*e + A*b^2*d^2*e^2 + (A*c^2*d^2*e^2 - 2*A*b*c*d*e^3 + A*b^2*e^4)*x^2 + 2*(A*c^2*d^3*e - 2*A*b*c*d^2*e^2 +
A*b^2*d*e^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(4*B*b*c*d^4 + 4*A*b^2*d^2*e^2 - (B*b
^2 + 7*A*b*c)*d^3*e + 3*(B*b*c*d^3*e - 2*A*b*c*d^2*e^2 + A*b^2*d*e^3)*x)*sqrt(e*x + d))/(b*c^2*d^7 - 2*b^2*c*d
^6*e + b^3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(b*c^2*d^6*e - 2*b^2*c*d^5*e^2 +
b^3*d^4*e^3)*x), -1/3*(6*((B*b*c - A*c^2)*d^3*e^2*x^2 + 2*(B*b*c - A*c^2)*d^4*e*x + (B*b*c - A*c^2)*d^5)*sqrt(
-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 3*(A*c^2*d^4 - 2*A*b*c
*d^3*e + A*b^2*d^2*e^2 + (A*c^2*d^2*e^2 - 2*A*b*c*d*e^3 + A*b^2*e^4)*x^2 + 2*(A*c^2*d^3*e - 2*A*b*c*d^2*e^2 +
A*b^2*d*e^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(4*B*b*c*d^4 + 4*A*b^2*d^2*e^2 - (B*b
^2 + 7*A*b*c)*d^3*e + 3*(B*b*c*d^3*e - 2*A*b*c*d^2*e^2 + A*b^2*d*e^3)*x)*sqrt(e*x + d))/(b*c^2*d^7 - 2*b^2*c*d
^6*e + b^3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(b*c^2*d^6*e - 2*b^2*c*d^5*e^2 +
b^3*d^4*e^3)*x), 1/3*(6*(A*c^2*d^4 - 2*A*b*c*d^3*e + A*b^2*d^2*e^2 + (A*c^2*d^2*e^2 - 2*A*b*c*d*e^3 + A*b^2*e^
4)*x^2 + 2*(A*c^2*d^3*e - 2*A*b*c*d^2*e^2 + A*b^2*d*e^3)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 3*((B*
b*c - A*c^2)*d^3*e^2*x^2 + 2*(B*b*c - A*c^2)*d^4*e*x + (B*b*c - A*c^2)*d^5)*sqrt(c/(c*d - b*e))*log((c*e*x + 2
*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(4*B*b*c*d^4 + 4*A*b^2*d^2*e^2 -
(B*b^2 + 7*A*b*c)*d^3*e + 3*(B*b*c*d^3*e - 2*A*b*c*d^2*e^2 + A*b^2*d*e^3)*x)*sqrt(e*x + d))/(b*c^2*d^7 - 2*b^2
*c*d^6*e + b^3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(b*c^2*d^6*e - 2*b^2*c*d^5*e^
2 + b^3*d^4*e^3)*x), -2/3*(3*((B*b*c - A*c^2)*d^3*e^2*x^2 + 2*(B*b*c - A*c^2)*d^4*e*x + (B*b*c - A*c^2)*d^5)*s
qrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 3*(A*c^2*d^4 - 2*A
*b*c*d^3*e + A*b^2*d^2*e^2 + (A*c^2*d^2*e^2 - 2*A*b*c*d*e^3 + A*b^2*e^4)*x^2 + 2*(A*c^2*d^3*e - 2*A*b*c*d^2*e^
2 + A*b^2*d*e^3)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (4*B*b*c*d^4 + 4*A*b^2*d^2*e^2 - (B*b^2 + 7*A*
b*c)*d^3*e + 3*(B*b*c*d^3*e - 2*A*b*c*d^2*e^2 + A*b^2*d*e^3)*x)*sqrt(e*x + d))/(b*c^2*d^7 - 2*b^2*c*d^6*e + b^
3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(b*c^2*d^6*e - 2*b^2*c*d^5*e^2 + b^3*d^4*e
^3)*x)]

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giac [A]  time = 0.22, size = 217, normalized size = 1.32 \begin {gather*} \frac {2 \, {\left (B b c^{2} - A c^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} \sqrt {-c^{2} d + b c e}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )} B c d^{2} + B c d^{3} - 6 \, {\left (x e + d\right )} A c d e - B b d^{2} e - A c d^{2} e + 3 \, {\left (x e + d\right )} A b e^{2} + A b d e^{2}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {2 \, A \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

2*(B*b*c^2 - A*c^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^2*d^2 - 2*b^2*c*d*e + b^3*e^2)*sqrt(-c^
2*d + b*c*e)) + 2/3*(3*(x*e + d)*B*c*d^2 + B*c*d^3 - 6*(x*e + d)*A*c*d*e - B*b*d^2*e - A*c*d^2*e + 3*(x*e + d)
*A*b*e^2 + A*b*d*e^2)/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*(x*e + d)^(3/2)) + 2*A*arctan(sqrt(x*e + d)/sqrt(
-d))/(b*sqrt(-d)*d^2)

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maple [A]  time = 0.07, size = 243, normalized size = 1.48 \begin {gather*} -\frac {2 A \,c^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}\, b}+\frac {2 B \,c^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {2 A b \,e^{2}}{\left (b e -c d \right )^{2} \sqrt {e x +d}\, d^{2}}-\frac {4 A c e}{\left (b e -c d \right )^{2} \sqrt {e x +d}\, d}+\frac {2 B c}{\left (b e -c d \right )^{2} \sqrt {e x +d}}+\frac {2 A e}{3 \left (b e -c d \right ) \left (e x +d \right )^{\frac {3}{2}} d}-\frac {2 B}{3 \left (b e -c d \right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 A \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(b*e-c*d)^2*c^3/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+2/(b*e-c*d)^2*c^2/((b*e
-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B-2*A*arctanh((e*x+d)^(1/2)/d^(1/2))/b/d^(5/2)+2/3/
(b*e-c*d)/d/(e*x+d)^(3/2)*A*e-2/3/(b*e-c*d)/(e*x+d)^(3/2)*B+2/(b*e-c*d)^2/d^2/(e*x+d)^(1/2)*A*b*e^2-4/(b*e-c*d
)^2/d/(e*x+d)^(1/2)*A*c*e+2/(b*e-c*d)^2/(e*x+d)^(1/2)*B*c

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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)/(c*x^2+b*x),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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 3.92, size = 6340, normalized size = 38.66

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^13*d^16*e^2 + 480*A^2*b^2*c^11*d^14*
e^4 - 1120*A^2*b^3*c^10*d^13*e^5 + 1800*A^2*b^4*c^9*d^12*e^6 - 2064*A^2*b^5*c^8*d^11*e^7 + 1688*A^2*b^6*c^7*d^
10*e^8 - 960*A^2*b^7*c^6*d^9*e^9 + 360*A^2*b^8*c^5*d^8*e^10 - 80*A^2*b^9*c^4*d^7*e^11 + 8*A^2*b^10*c^3*d^6*e^1
2 + 8*B^2*b^2*c^11*d^16*e^2 - 48*B^2*b^3*c^10*d^15*e^3 + 120*B^2*b^4*c^9*d^14*e^4 - 160*B^2*b^5*c^8*d^13*e^5 +
120*B^2*b^6*c^7*d^12*e^6 - 48*B^2*b^7*c^6*d^11*e^7 + 8*B^2*b^8*c^5*d^10*e^8 - 128*A^2*b*c^12*d^15*e^3 - 16*A*
B*b*c^12*d^16*e^2 + 96*A*B*b^2*c^11*d^15*e^3 - 240*A*B*b^3*c^10*d^14*e^4 + 320*A*B*b^4*c^9*d^13*e^5 - 240*A*B*
b^5*c^8*d^12*e^6 + 96*A*B*b^6*c^7*d^11*e^7 - 16*A*B*b^7*c^6*d^10*e^8) - ((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b
)*(((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 800
*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^1
5*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*b
^13*c^2*d^10*e^13))/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c
*d*e^4) - 24*A*b^2*c^12*d^18*e^3 + 216*A*b^3*c^11*d^17*e^4 - 872*A*b^4*c^10*d^16*e^5 + 2080*A*b^5*c^9*d^15*e^6
- 3248*A*b^6*c^8*d^14*e^7 + 3472*A*b^7*c^7*d^13*e^8 - 2576*A*b^8*c^6*d^12*e^9 + 1312*A*b^9*c^5*d^11*e^10 - 44
0*A*b^10*c^4*d^10*e^11 + 88*A*b^11*c^3*d^9*e^12 - 8*A*b^12*c^2*d^8*e^13 + 8*B*b^2*c^12*d^19*e^2 - 64*B*b^3*c^1
1*d^18*e^3 + 224*B*b^4*c^10*d^17*e^4 - 448*B*b^5*c^9*d^16*e^5 + 560*B*b^6*c^8*d^15*e^6 - 448*B*b^7*c^7*d^14*e^
7 + 224*B*b^8*c^6*d^13*e^8 - 64*B*b^9*c^5*d^12*e^9 + 8*B*b^10*c^4*d^11*e^10))/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4
*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4))*1i)/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e
- 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4) + ((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*((d + e*x
)^(1/2)*(16*A^2*c^13*d^16*e^2 + 480*A^2*b^2*c^11*d^14*e^4 - 1120*A^2*b^3*c^10*d^13*e^5 + 1800*A^2*b^4*c^9*d^12
*e^6 - 2064*A^2*b^5*c^8*d^11*e^7 + 1688*A^2*b^6*c^7*d^10*e^8 - 960*A^2*b^7*c^6*d^9*e^9 + 360*A^2*b^8*c^5*d^8*e
^10 - 80*A^2*b^9*c^4*d^7*e^11 + 8*A^2*b^10*c^3*d^6*e^12 + 8*B^2*b^2*c^11*d^16*e^2 - 48*B^2*b^3*c^10*d^15*e^3 +
120*B^2*b^4*c^9*d^14*e^4 - 160*B^2*b^5*c^8*d^13*e^5 + 120*B^2*b^6*c^7*d^12*e^6 - 48*B^2*b^7*c^6*d^11*e^7 + 8*
B^2*b^8*c^5*d^10*e^8 - 128*A^2*b*c^12*d^15*e^3 - 16*A*B*b*c^12*d^16*e^2 + 96*A*B*b^2*c^11*d^15*e^3 - 240*A*B*b
^3*c^10*d^14*e^4 + 320*A*B*b^4*c^9*d^13*e^5 - 240*A*B*b^5*c^8*d^12*e^6 + 96*A*B*b^6*c^7*d^11*e^7 - 16*A*B*b^7*
c^6*d^10*e^8) - ((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*(((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*(d + e*x)^(1/
2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b^6*c
^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e^10
- 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*b^13*c^2*d^10*e^13))/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4
*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4) + 24*A*b^2*c^12*d^18*e^3 - 216*A*b^3*c^11*d^17*e
^4 + 872*A*b^4*c^10*d^16*e^5 - 2080*A*b^5*c^9*d^15*e^6 + 3248*A*b^6*c^8*d^14*e^7 - 3472*A*b^7*c^7*d^13*e^8 + 2
576*A*b^8*c^6*d^12*e^9 - 1312*A*b^9*c^5*d^11*e^10 + 440*A*b^10*c^4*d^10*e^11 - 88*A*b^11*c^3*d^9*e^12 + 8*A*b^
12*c^2*d^8*e^13 - 8*B*b^2*c^12*d^19*e^2 + 64*B*b^3*c^11*d^18*e^3 - 224*B*b^4*c^10*d^17*e^4 + 448*B*b^5*c^9*d^1
6*e^5 - 560*B*b^6*c^8*d^15*e^6 + 448*B*b^7*c^7*d^14*e^7 - 224*B*b^8*c^6*d^13*e^8 + 64*B*b^9*c^5*d^12*e^9 - 8*B
*b^10*c^4*d^11*e^10))/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5
*c*d*e^4))*1i)/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^
4))/(32*A^3*c^12*d^13*e^3 + ((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^13*d^16*e^2 + 4
80*A^2*b^2*c^11*d^14*e^4 - 1120*A^2*b^3*c^10*d^13*e^5 + 1800*A^2*b^4*c^9*d^12*e^6 - 2064*A^2*b^5*c^8*d^11*e^7
+ 1688*A^2*b^6*c^7*d^10*e^8 - 960*A^2*b^7*c^6*d^9*e^9 + 360*A^2*b^8*c^5*d^8*e^10 - 80*A^2*b^9*c^4*d^7*e^11 + 8
*A^2*b^10*c^3*d^6*e^12 + 8*B^2*b^2*c^11*d^16*e^2 - 48*B^2*b^3*c^10*d^15*e^3 + 120*B^2*b^4*c^9*d^14*e^4 - 160*B
^2*b^5*c^8*d^13*e^5 + 120*B^2*b^6*c^7*d^12*e^6 - 48*B^2*b^7*c^6*d^11*e^7 + 8*B^2*b^8*c^5*d^10*e^8 - 128*A^2*b*
c^12*d^15*e^3 - 16*A*B*b*c^12*d^16*e^2 + 96*A*B*b^2*c^11*d^15*e^3 - 240*A*B*b^3*c^10*d^14*e^4 + 320*A*B*b^4*c^
9*d^13*e^5 - 240*A*B*b^5*c^8*d^12*e^6 + 96*A*B*b^6*c^7*d^11*e^7 - 16*A*B*b^7*c^6*d^10*e^8) - ((-c^3*(b*e - c*d
)^5)^(1/2)*(A*c - B*b)*(((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^
3*c^12*d^20*e^3 + 800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e
^7 + 5376*b^8*c^7*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^1
2*c^3*d^11*e^12 - 8*b^13*c^2*d^10*e^13))/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*
c^2*d^2*e^3 - 5*b^5*c*d*e^4) - 24*A*b^2*c^12*d^18*e^3 + 216*A*b^3*c^11*d^17*e^4 - 872*A*b^4*c^10*d^16*e^5 + 20
80*A*b^5*c^9*d^15*e^6 - 3248*A*b^6*c^8*d^14*e^7 + 3472*A*b^7*c^7*d^13*e^8 - 2576*A*b^8*c^6*d^12*e^9 + 1312*A*b
^9*c^5*d^11*e^10 - 440*A*b^10*c^4*d^10*e^11 + 88*A*b^11*c^3*d^9*e^12 - 8*A*b^12*c^2*d^8*e^13 + 8*B*b^2*c^12*d^
19*e^2 - 64*B*b^3*c^11*d^18*e^3 + 224*B*b^4*c^10*d^17*e^4 - 448*B*b^5*c^9*d^16*e^5 + 560*B*b^6*c^8*d^15*e^6 -
448*B*b^7*c^7*d^14*e^7 + 224*B*b^8*c^6*d^13*e^8 - 64*B*b^9*c^5*d^12*e^9 + 8*B*b^10*c^4*d^11*e^10))/(b^6*e^5 -
b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4)))/(b^6*e^5 - b*c^5*d^5
+ 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4) - ((-c^3*(b*e - c*d)^5)^(1/2)*(A*
c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^13*d^16*e^2 + 480*A^2*b^2*c^11*d^14*e^4 - 1120*A^2*b^3*c^10*d^13*e^5 + 180
0*A^2*b^4*c^9*d^12*e^6 - 2064*A^2*b^5*c^8*d^11*e^7 + 1688*A^2*b^6*c^7*d^10*e^8 - 960*A^2*b^7*c^6*d^9*e^9 + 360
*A^2*b^8*c^5*d^8*e^10 - 80*A^2*b^9*c^4*d^7*e^11 + 8*A^2*b^10*c^3*d^6*e^12 + 8*B^2*b^2*c^11*d^16*e^2 - 48*B^2*b
^3*c^10*d^15*e^3 + 120*B^2*b^4*c^9*d^14*e^4 - 160*B^2*b^5*c^8*d^13*e^5 + 120*B^2*b^6*c^7*d^12*e^6 - 48*B^2*b^7
*c^6*d^11*e^7 + 8*B^2*b^8*c^5*d^10*e^8 - 128*A^2*b*c^12*d^15*e^3 - 16*A*B*b*c^12*d^16*e^2 + 96*A*B*b^2*c^11*d^
15*e^3 - 240*A*B*b^3*c^10*d^14*e^4 + 320*A*B*b^4*c^9*d^13*e^5 - 240*A*B*b^5*c^8*d^12*e^6 + 96*A*B*b^6*c^7*d^11
*e^7 - 16*A*B*b^7*c^6*d^10*e^8) - ((-c^3*(b*e - c*d)^5)^(1/2)*(A*c - B*b)*(((-c^3*(b*e - c*d)^5)^(1/2)*(A*c -
B*b)*(d + e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^1
8*e^5 + 4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b
^10*c^5*d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*b^13*c^2*d^10*e^13))/(b^6*e^5 - b*c^5*d
^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4) + 24*A*b^2*c^12*d^18*e^3 - 216
*A*b^3*c^11*d^17*e^4 + 872*A*b^4*c^10*d^16*e^5 - 2080*A*b^5*c^9*d^15*e^6 + 3248*A*b^6*c^8*d^14*e^7 - 3472*A*b^
7*c^7*d^13*e^8 + 2576*A*b^8*c^6*d^12*e^9 - 1312*A*b^9*c^5*d^11*e^10 + 440*A*b^10*c^4*d^10*e^11 - 88*A*b^11*c^3
*d^9*e^12 + 8*A*b^12*c^2*d^8*e^13 - 8*B*b^2*c^12*d^19*e^2 + 64*B*b^3*c^11*d^18*e^3 - 224*B*b^4*c^10*d^17*e^4 +
448*B*b^5*c^9*d^16*e^5 - 560*B*b^6*c^8*d^15*e^6 + 448*B*b^7*c^7*d^14*e^7 - 224*B*b^8*c^6*d^13*e^8 + 64*B*b^9*
c^5*d^12*e^9 - 8*B*b^10*c^4*d^11*e^10))/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c
^2*d^2*e^3 - 5*b^5*c*d*e^4)))/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d^2*e^3
- 5*b^5*c*d*e^4) + 576*A^3*b^2*c^10*d^11*e^5 - 880*A^3*b^3*c^9*d^10*e^6 + 800*A^3*b^4*c^8*d^9*e^7 - 432*A^3*b
^5*c^7*d^8*e^8 + 128*A^3*b^6*c^6*d^7*e^9 - 16*A^3*b^7*c^5*d^6*e^10 - 16*A^2*B*c^12*d^14*e^2 - 208*A^3*b*c^11*d
^12*e^4 - 96*A*B^2*b^2*c^10*d^13*e^3 + 240*A*B^2*b^3*c^9*d^12*e^4 - 320*A*B^2*b^4*c^8*d^11*e^5 + 240*A*B^2*b^5
*c^7*d^10*e^6 - 96*A*B^2*b^6*c^6*d^9*e^7 + 16*A*B^2*b^7*c^5*d^8*e^8 - 32*A^2*B*b^2*c^10*d^12*e^4 - 256*A^2*B*b
^3*c^9*d^11*e^5 + 640*A^2*B*b^4*c^8*d^10*e^6 - 704*A^2*B*b^5*c^7*d^9*e^7 + 416*A^2*B*b^6*c^6*d^8*e^8 - 128*A^2
*B*b^7*c^5*d^7*e^9 + 16*A^2*B*b^8*c^4*d^6*e^10 + 16*A*B^2*b*c^11*d^14*e^2 + 64*A^2*B*b*c^11*d^13*e^3))*(-c^3*(
b*e - c*d)^5)^(1/2)*(A*c - B*b)*2i)/(b^6*e^5 - b*c^5*d^5 + 5*b^2*c^4*d^4*e - 10*b^3*c^3*d^3*e^2 + 10*b^4*c^2*d
^2*e^3 - 5*b^5*c*d*e^4) - (A*atan((B^2*b^2*c^9*d^21*(d + e*x)^(1/2)*1i + A^2*b^11*d^10*e^11*(d + e*x)^(1/2)*1i
- A^2*b^10*c*d^11*e^10*(d + e*x)^(1/2)*11i - B^2*b^3*c^8*d^20*e*(d + e*x)^(1/2)*6i - A*B*b*c^10*d^21*(d + e*x
)^(1/2)*2i - A^2*b^2*c^9*d^19*e^2*(d + e*x)^(1/2)*40i + A^2*b^3*c^8*d^18*e^3*(d + e*x)^(1/2)*145i - A^2*b^4*c^
7*d^17*e^4*(d + e*x)^(1/2)*315i + A^2*b^5*c^6*d^16*e^5*(d + e*x)^(1/2)*456i - A^2*b^6*c^5*d^15*e^6*(d + e*x)^(
1/2)*461i + A^2*b^7*c^4*d^14*e^7*(d + e*x)^(1/2)*330i - A^2*b^8*c^3*d^13*e^8*(d + e*x)^(1/2)*165i + A^2*b^9*c^
2*d^12*e^9*(d + e*x)^(1/2)*55i + B^2*b^4*c^7*d^19*e^2*(d + e*x)^(1/2)*15i - B^2*b^5*c^6*d^18*e^3*(d + e*x)^(1/
2)*20i + B^2*b^6*c^5*d^17*e^4*(d + e*x)^(1/2)*15i - B^2*b^7*c^4*d^16*e^5*(d + e*x)^(1/2)*6i + B^2*b^8*c^3*d^15
*e^6*(d + e*x)^(1/2)*1i + A^2*b*c^10*d^20*e*(d + e*x)^(1/2)*5i - A*B*b^3*c^8*d^19*e^2*(d + e*x)^(1/2)*30i + A*
B*b^4*c^7*d^18*e^3*(d + e*x)^(1/2)*40i - A*B*b^5*c^6*d^17*e^4*(d + e*x)^(1/2)*30i + A*B*b^6*c^5*d^16*e^5*(d +
e*x)^(1/2)*12i - A*B*b^7*c^4*d^15*e^6*(d + e*x)^(1/2)*2i + A*B*b^2*c^9*d^20*e*(d + e*x)^(1/2)*12i)/(d^5*(d^5)^
(1/2)*(d^5*(d^5*(315*A^2*b^4*c^7*e^4 - B^2*b^2*c^9*d^4 - 15*B^2*b^6*c^5*e^4 + 40*A^2*b^2*c^9*d^2*e^2 - 15*B^2*
b^4*c^7*d^2*e^2 + 30*A*B*b^5*c^6*e^4 - 5*A^2*b*c^10*d^3*e - 145*A^2*b^3*c^8*d*e^3 + 6*B^2*b^3*c^8*d^3*e + 20*B
^2*b^5*c^6*d*e^3 + 2*A*B*b*c^10*d^4 - 12*A*B*b^2*c^9*d^3*e - 40*A*B*b^4*c^7*d*e^3 + 30*A*B*b^3*c^8*d^2*e^2) -
55*A^2*b^9*c^2*e^9 - 456*A^2*b^5*c^6*d^4*e^5 + 461*A^2*b^6*c^5*d^3*e^6 - 330*A^2*b^7*c^4*d^2*e^7 + 6*B^2*b^7*c
^4*d^4*e^5 - B^2*b^8*c^3*d^3*e^6 + 165*A^2*b^8*c^3*d*e^8 - 12*A*B*b^6*c^5*d^4*e^5 + 2*A*B*b^7*c^4*d^3*e^6) - A
^2*b^11*d^3*e^11 + 11*A^2*b^10*c*d^4*e^10)))*2i)/(b*(d^5)^(1/2)) - ((2*(A*e - B*d))/(3*(c*d^2 - b*d*e)) - (2*(
d + e*x)*(A*b*e^2 + B*c*d^2 - 2*A*c*d*e))/(c*d^2 - b*d*e)^2)/(d + e*x)^(3/2)

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sympy [A]  time = 66.52, size = 160, normalized size = 0.98 \begin {gather*} \frac {2 A \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d^{2} \sqrt {- d}} - \frac {2 \left (- A e + B d\right )}{3 d \left (d + e x\right )^{\frac {3}{2}} \left (b e - c d\right )} + \frac {2 \left (A b e^{2} - 2 A c d e + B c d^{2}\right )}{d^{2} \sqrt {d + e x} \left (b e - c d\right )^{2}} + \frac {2 c \left (- A c + B b\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b \sqrt {\frac {b e - c d}{c}} \left (b e - c d\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(5/2)/(c*x**2+b*x),x)

[Out]

2*A*atan(sqrt(d + e*x)/sqrt(-d))/(b*d**2*sqrt(-d)) - 2*(-A*e + B*d)/(3*d*(d + e*x)**(3/2)*(b*e - c*d)) + 2*(A*
b*e**2 - 2*A*c*d*e + B*c*d**2)/(d**2*sqrt(d + e*x)*(b*e - c*d)**2) + 2*c*(-A*c + B*b)*atan(sqrt(d + e*x)/sqrt(
(b*e - c*d)/c))/(b*sqrt((b*e - c*d)/c)*(b*e - c*d)**2)

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