3.420 \(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=478 \[ -\frac {4 e \sqrt {b x+c x^2} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {4 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}-\frac {2 e \sqrt {b x+c x^2} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]

[Out]

-2*(b*(-b*e+c*d)+c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(3/2)/(c*x^2+b*x)^(1/2)+2/3*(-b*e+2*c*d)*(8*b^2*e^
2-3*b*c*d*e+3*c^2*d^2)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(
e*x+d)^(1/2)/(-b)^(3/2)/d^3/(-b*e+c*d)^3/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-4/3*(2*b^2*e^2-3*b*c*d*e+3*c^2*d^2)
*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(3
/2)/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-4/3*e*(2*b^2*e^2-3*b*c*d*e+3*c^2*d^2)*(c*x^2+b*x)^(1/2)/b
^2/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)-2/3*e*(-b*e+2*c*d)*(8*b^2*e^2-3*b*c*d*e+3*c^2*d^2)*(c*x^2+b*x)^(1/2)/b^2/d^3
/(-b*e+c*d)^3/(e*x+d)^(1/2)

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Rubi [A]  time = 0.59, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {740, 834, 843, 715, 112, 110, 117, 116} \[ -\frac {4 e \sqrt {b x+c x^2} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 e \sqrt {b x+c x^2} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {4 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2]) - (4*e*(3*c^2*d
^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[b*x + c*x^2])/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (2*e*(2*c*d - b*e)*
(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[b*x + c*x^2])/(3*b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) + (2*Sqrt[c]*(2
*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt
[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
4*Sqrt[c]*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sq
rt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, x] /; FreeQ[{b, c, d, e, f}, x] &&
NeQ[d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-(b/d), 0]

Rule 112

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*x]*Sqrt[
1 + (d*x)/c])/(Sqrt[c + d*x]*Sqrt[1 + (f*x)/e]), Int[Sqrt[1 + (f*x)/e]/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 715

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(Sqrt[x]*Sqrt[b + c*x])/Sqrt[
b*x + c*x^2], Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 843

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} b e (3 c d-4 b e)+\frac {3}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {4 \int \frac {-\frac {1}{4} b e \left (3 c^2 d^2-15 b c d e+8 b^2 e^2\right )-\frac {1}{2} c e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{3 b^2 d^2 (c d-b e)^2}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {1}{8} b c d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )-\frac {1}{8} c e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {\left (2 c \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 d^2 (c d-b e)^2}+\frac {\left (c (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {\left (2 c \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^2 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^2 d^3 (c d-b e)^3 \sqrt {b x+c x^2}}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\left (c (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 b^2 d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (2 c \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 b^2 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}-\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {2 \sqrt {c} (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 1.15, size = 420, normalized size = 0.88 \[ -\frac {2 \left (b \left (b^2 d e^3 x (b+c x) (c d-b e)-5 b^2 e^3 x (b+c x) (d+e x) (b e-2 c d)+3 (b+c x) (d+e x)^2 (c d-b e)^3+3 c^4 d^3 x (d+e x)^2\right )-c \sqrt {\frac {b}{c}} (d+e x) \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{3 b^3 d^3 \sqrt {x (b+c x)} (d+e x)^{3/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*(b^2*d*e^3*(c*d - b*e)*x*(b + c*x) - 5*b^2*e^3*(-2*c*d + b*e)*x*(b + c*x)*(d + e*x) + 3*c^4*d^3*x*(d +
e*x)^2 + 3*(c*d - b*e)^3*(b + c*x)*(d + e*x)^2) - Sqrt[b/c]*c*(d + e*x)*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e
+ 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*
e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*
(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elliptic
F[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^3*d^3*(c*d - b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))

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fricas [F]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{2} e^{3} x^{7} + b^{2} d^{3} x^{2} + {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{6} + {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{5} + {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{4} + {\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(c^2*e^3*x^7 + b^2*d^3*x^2 + (3*c^2*d*e^2 + 2*b*c*e^3)*x^6 + (3*c^2*d
^2*e + 6*b*c*d*e^2 + b^2*e^3)*x^5 + (c^2*d^3 + 6*b*c*d^2*e + 3*b^2*d*e^2)*x^4 + (2*b*c*d^3 + 3*b^2*d^2*e)*x^3)
, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
3.26Unable to transpose Error: Bad Argument Value

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maple [B]  time = 0.19, size = 1708, normalized size = 3.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x)

[Out]

-2/3*((c*x+b)*x)^(1/2)/x*(-20*b^2*c^3*d^2*e^3*x^2+15*b*c^4*d^3*e^2*x^2-26*b^3*c^2*d^2*e^3*x+9*b^2*c^3*d^3*e^2*
x+3*b*c^4*d^4*e*x+8*x^2*b^4*c*e^5+12*x*b^4*c*d*e^4+12*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b
^2*c^3*d^3*e^2*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-6*EllipticF(((c*x+b)/b)^(1/2),(
1/(b*e-c*d)*b*e)^(1/2))*x*b*c^4*d^4*e*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-27*Ellip
ticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*c*d*e^4*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((
c*x+b)/b)^(1/2)+28*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c^2*d^2*e^3*(-(e*x+d)/(b*e-c*d)*
c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-15*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c^3*
d^3*e^2*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)+6*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-
c*d)*b*e)^(1/2))*x*b*c^4*d^4*e*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)+4*EllipticF(((c
*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*c*d*e^4*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b
)^(1/2)-10*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c^2*d^2*e^3*(-(e*x+d)/(b*e-c*d)*c)^(1/2)
*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-15*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^3*d^4*e*(-(e
*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-6*x*c^5*d^5+3*b^4*c*d^2*e^3-9*b^3*c^2*d^3*e^2+9*b^
2*c^3*d^4*e+8*b^3*c^2*e^5*x^3-6*c^5*d^3*e^2*x^3-12*c^5*d^4*e*x^2+4*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*
e)^(1/2))*b^4*c*d^2*e^3*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-10*EllipticF(((c*x+b)/
b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*c^2*d^3*e^2*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1
/2)+12*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^3*d^4*e*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c
*x)^(1/2)*((c*x+b)/b)^(1/2)-27*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*c*d^2*e^3*(-(e*x+d)/(b
*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)+28*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^
3*c^2*d^3*e^2*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-19*b^2*c^3*d*e^4*x^3+9*b*c^4*d^2
*e^3*x^3-7*b^3*c^2*d*e^4*x^2-3*b*c^4*d^5+8*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^5*d*e^4*(-(e
*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)+6*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1
/2))*b*c^4*d^5*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)+8*EllipticE(((c*x+b)/b)^(1/2),(
1/(b*e-c*d)*b*e)^(1/2))*x*b^5*e^5*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)^(1/2)-6*EllipticF(
((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^4*d^5*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*((c*x+b)/b)
^(1/2))/d^3/b^2/c/(e*x+d)^(3/2)/(b*e-c*d)^3/(c*x+b)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^(5/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^(3/2)*(d + e*x)^(5/2)),x)

[Out]

int(1/((b*x + c*x^2)^(3/2)*(d + e*x)^(5/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(1/((x*(b + c*x))**(3/2)*(d + e*x)**(5/2)), x)

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