3.1270 \(\int \frac{A+B x}{(d+e x)^{7/2} \sqrt{b x+c x^2}} \, dx\)
Optimal. Leaf size=510 \[ -\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 A e (2 c d-b e)-B d (b e+3 c d)) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{15 d^2 e \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 \sqrt{b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{15 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^3 e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 \sqrt{b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}+\frac{2 \sqrt{b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)} \]
[Out]
(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(4*A*e*(2*c*d - b*e) - B*d*(3*c*d + b
*e))*Sqrt[b*x + c*x^2])/(15*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) + (2*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) -
A*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2))*Sqrt[b*x + c*x^2])/(15*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (2*Sqrt[
-b]*Sqrt[c]*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e*(23*c^2*d^2 - 23*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)])/(15*d^3*e*(c*d - b*e)^
3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(4*A*e*(2*c*d - b*e) - B*d*(3*c*d + b*e))*Sqrt[x]
*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^2*e*(c*
d - b*e)^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi [A] time = 0.800803, antiderivative size = 510, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used =
{834, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{15 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^3 e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 \sqrt{b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 A e (2 c d-b e)-B d (b e+3 c d)) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 e \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 \sqrt{b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(7/2)*Sqrt[b*x + c*x^2]),x]
[Out]
(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(4*A*e*(2*c*d - b*e) - B*d*(3*c*d + b
*e))*Sqrt[b*x + c*x^2])/(15*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) + (2*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) -
A*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2))*Sqrt[b*x + c*x^2])/(15*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (2*Sqrt[
-b]*Sqrt[c]*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e*(23*c^2*d^2 - 23*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)])/(15*d^3*e*(c*d - b*e)^
3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(4*A*e*(2*c*d - b*e) - B*d*(3*c*d + b*e))*Sqrt[x]
*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^2*e*(c*
d - b*e)^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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]
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 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 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 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 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)])
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^{7/2} \sqrt{b x+c x^2}} \, dx &=\frac{2 (B d-A e) \sqrt{b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac{2 \int \frac{\frac{1}{2} (b B d-5 A c d+4 A b e)-\frac{3}{2} c (B d-A e) x}{(d+e x)^{5/2} \sqrt{b x+c x^2}} \, dx}{5 d (c d-b e)}\\ &=\frac{2 (B d-A e) \sqrt{b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac{2 (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt{b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{4 \int \frac{\frac{1}{4} \left (15 A c^2 d^2+2 b^2 e (B d+4 A e)-b c d (6 B d+19 A e)\right )-\frac{1}{4} c (4 A e (2 c d-b e)-B d (3 c d+b e)) x}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx}{15 d^2 (c d-b e)^2}\\ &=\frac{2 (B d-A e) \sqrt{b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac{2 (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt{b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{2 \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{8 \int \frac{-\frac{1}{8} c d \left (15 A c^2 d^2+b^2 e (B d+4 A e)-b c d (9 B d+11 A e)\right )+\frac{1}{8} c \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 d^3 (c d-b e)^3}\\ &=\frac{2 (B d-A e) \sqrt{b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac{2 (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt{b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{2 \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{(c (4 A e (2 c d-b e)-B d (3 c d+b e))) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 d^2 e (c d-b e)^2}-\frac{\left (c \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{15 d^3 e (c d-b e)^3}\\ &=\frac{2 (B d-A e) \sqrt{b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac{2 (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt{b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{2 \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{\left (c (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{15 d^2 e (c d-b e)^2 \sqrt{b x+c x^2}}-\frac{\left (c \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{15 d^3 e (c d-b e)^3 \sqrt{b x+c x^2}}\\ &=\frac{2 (B d-A e) \sqrt{b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac{2 (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt{b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{2 \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{\left (c \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\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}{15 d^3 e (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (c (4 A e (2 c d-b e)-B d (3 c d+b e)) \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}{15 d^2 e (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 (B d-A e) \sqrt{b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac{2 (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt{b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{2 \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt{b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{c} \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\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 )}{15 d^3 e (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} \sqrt{c} (4 A e (2 c d-b e)-B d (3 c d+b e)) \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 )}{15 d^2 e (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 3.06871, size = 506, normalized size = 0.99 \[ \frac{2 \left (b e x (b+c x) \left ((d+e x)^2 \left (A e \left (-8 b^2 e^2+23 b c d e-23 c^2 d^2\right )+B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )\right )+3 d^2 (B d-A e) (c d-b e)^2+d (d+e x) (c d-b e) (4 A e (b e-2 c d)+B d (b e+3 c d))\right )-c \sqrt{\frac{b}{c}} (d+e x)^2 \left (-i e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (2 b^2 e (4 A e+B d)-b c d (19 A e+6 B d)+15 A c^2 d^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+B d \left (2 b^2 e^2-7 b c d e-3 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (A e \left (-8 b^2 e^2+23 b c d e-23 c^2 d^2\right )+B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )\right )\right )\right )}{15 b d^3 e \sqrt{x (b+c x)} (d+e x)^{5/2} (c d-b e)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(7/2)*Sqrt[b*x + c*x^2]),x]
[Out]
(2*(b*e*x*(b + c*x)*(3*d^2*(B*d - A*e)*(c*d - b*e)^2 + d*(c*d - b*e)*(4*A*e*(-2*c*d + b*e) + B*d*(3*c*d + b*e)
)*(d + e*x) + (A*e*(-23*c^2*d^2 + 23*b*c*d*e - 8*b^2*e^2) + B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2))*(d + e*x)
^2) - Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(A*e*(-23*c^2*d^2 + 23*b*c*d*e - 8*b^2*e^2) + B*d*(3*c^2*d^2 + 7*b*c*
d*e - 2*b^2*e^2))*(b + c*x)*(d + e*x) - I*b*e*(B*d*(-3*c^2*d^2 - 7*b*c*d*e + 2*b^2*e^2) + A*e*(23*c^2*d^2 - 23
*b*c*d*e + 8*b^2*e^2))*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*e*(c*d - b*e)*(15*A*c^2*d^2 + 2*b^2*e*(B*d + 4*A*e) - b*c*d*(6*B*d + 19*A*e))*Sqrt[1 + b/(c*x)]*
Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b*d^3*e*(c*d - b*e)^3*Sq
rt[x*(b + c*x)]*(d + e*x)^(5/2))
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Maple [B] time = 0.047, size = 3863, normalized size = 7.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x)
[Out]
2/15*(x*(c*x+b))^(1/2)*(-23*A*x^4*b*c^3*d*e^5+8*A*x^4*b^2*c^2*e^6+5*B*x^2*b^3*c*d^2*e^4+34*A*x*b*c^3*d^4*e^2+B
*x*b^2*c^2*d^4*e^2-9*B*x*b*c^3*d^5*e+2*B*x^4*b^2*c^2*d*e^5-7*B*x^4*b*c^3*d^2*e^4-3*A*x^3*b^2*c^2*d*e^5-35*A*x^
3*b*c^3*d^2*e^4+2*B*x^3*b^3*c*d*e^5-2*B*x^3*b^2*c^2*d^2*e^4-15*B*x^3*b*c^3*d^3*e^3+20*A*x^2*b^3*c*d*e^5-43*A*x
^2*b^2*c^2*d^2*e^4+13*A*x^2*b*c^3*d^3*e^3-12*B*x^2*b^2*c^2*d^3*e^3-8*B*x^2*b*c^3*d^4*e^2-41*A*x*b^2*c^2*d^3*e^
3+15*A*x*b^3*c*d^2*e^4+23*A*x^4*c^4*d^2*e^4-3*B*x^4*c^4*d^3*e^3+8*A*x^3*b^3*c*e^6+54*A*x^3*c^4*d^3*e^3-9*B*x^3
*c^4*d^4*e^2+34*A*x^2*c^4*d^4*e^2-9*B*x^2*c^4*d^5*e+3*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2
*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+B*EllipticF(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*x^2*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*B*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)-3*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-
(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-62*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^2*
e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+92*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-46*A*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^
(1/2)+8*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*
e-c*d))^(1/2)*(-c*x/b)^(1/2)-24*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b
)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+16*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*x*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-18*B*EllipticE(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*B*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)+6*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*
x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^3*e^3*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*x*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6*B*EllipticF(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-31
*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))
^(1/2)*(-c*x/b)^(1/2)+46*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-23*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*
b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*A*EllipticF(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-12*A*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+8*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-
(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-9*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d^2
*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*B*EllipticE(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)+16*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*
d))^(1/2)*(-c*x/b)^(1/2)+4*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d^2*e^4*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-31*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^
3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+46*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e
-c*d))^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-23*A*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)+4*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d)
)^(1/2)*(-c*x/b)^(1/2)-12*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^
5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-9*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*b^3*c*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*B*EllipticE(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+B*
EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+2*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*e^6*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2)
)*b^4*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*B*EllipticE(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*b^4*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*B*EllipticE((
(c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)-3*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1
/2)*(-c*x/b)^(1/2))/(c*x+b)/x/(b*e-c*d)^3/(e*x+d)^(5/2)/e/c/d^3
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x}{\left (B x + A\right )} \sqrt{e x + d}}{c e^{4} x^{6} + b d^{4} x +{\left (4 \, c d e^{3} + b e^{4}\right )} x^{5} + 2 \,{\left (3 \, c d^{2} e^{2} + 2 \, b d e^{3}\right )} x^{4} + 2 \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2}\right )} x^{3} +{\left (c d^{4} + 4 \, b d^{3} e\right )} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(c*x^2 + b*x)*(B*x + A)*sqrt(e*x + d)/(c*e^4*x^6 + b*d^4*x + (4*c*d*e^3 + b*e^4)*x^5 + 2*(3*c*d^2
*e^2 + 2*b*d*e^3)*x^4 + 2*(2*c*d^3*e + 3*b*d^2*e^2)*x^3 + (c*d^4 + 4*b*d^3*e)*x^2), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")
[Out]
integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)), x)