3.1270 \(\int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx\)

Optimal. Leaf size=510 \[ \frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{15 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^3 e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (4 A e (2 c d-b e)-B d (b e+3 c d)) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)} \]

[Out]

-2/15*(B*d*(-2*b^2*e^2+7*b*c*d*e+3*c^2*d^2)-A*e*(8*b^2*e^2-23*b*c*d*e+23*c^2*d^2))*EllipticE(c^(1/2)*x^(1/2)/(
-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/d^3/e/(-b*e+c*d)^3/(1+e*x/
d)^(1/2)/(c*x^2+b*x)^(1/2)-2/15*(4*A*e*(-b*e+2*c*d)-B*d*(b*e+3*c*d))*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e
/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1/2)/d^2/e/(-b*e+c*d)^2/(e*x+d)^(1/2)/(c*x^
2+b*x)^(1/2)+2/5*(-A*e+B*d)*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^(5/2)-2/15*(4*A*e*(-b*e+2*c*d)-B*d*(b*e+3*c
*d))*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)+2/15*(B*d*(-2*b^2*e^2+7*b*c*d*e+3*c^2*d^2)-A*e*(8*b^2*e^
2-23*b*c*d*e+23*c^2*d^2))*(c*x^2+b*x)^(1/2)/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)

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Rubi [A]  time = 0.80, antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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 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 {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*}

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Mathematica [C]  time = 3.03, size = 506, normalized size = 0.99 \[ \frac {2 \left (b e x (b+c x) \left ((d+e x)^2 \left (A e \left (-8 b^2 e^2+23 b c d e-23 c^2 d^2\right )+B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )\right )+3 d^2 (B d-A e) (c d-b e)^2+d (d+e x) (c d-b e) (4 A e (b e-2 c d)+B d (b e+3 c d))\right )-c \sqrt {\frac {b}{c}} (d+e x)^2 \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+B d \left (2 b^2 e^2-7 b c d e-3 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (A e \left (-8 b^2 e^2+23 b c d e-23 c^2 d^2\right )+B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )\right )-i e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (c d-b e) \left (2 b^2 e (4 A e+B d)-b c d (19 A e+6 B d)+15 A c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{15 b d^3 e \sqrt {x (b+c x)} (d+e x)^{5/2} (c d-b e)^3} \]

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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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{c e^{4} x^{6} + b d^{4} x + {\left (4 \, c d e^{3} + b e^{4}\right )} x^{5} + 2 \, {\left (3 \, c d^{2} e^{2} + 2 \, b d e^{3}\right )} x^{4} + 2 \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2}\right )} x^{3} + {\left (c d^{4} + 4 \, b d^{3} e\right )} x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]

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)

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maple [B]  time = 0.14, size = 3863, normalized size = 7.57 \[ \text {output too large to display} \]

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*((c*x+b)*x)^(1/2)*(-23*A*x^4*b*c^3*d*e^5+2*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^4
*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+16*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*
e-c*d)*b*e)^(1/2))*x*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+2*B*EllipticF((
(c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c
*x)^(1/2)+4*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(
b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-31*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*c*d^3*e^3*((c
*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+46*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e
)^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-23*A*EllipticE(((c*x+
b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/
2)+34*A*x^2*c^4*d^4*e^2-62*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/b)^
(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+4*A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b
^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-12*A*EllipticF(((c*x+b)/b)^(1/2),
(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+8*A*E
llipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)
*(-1/b*c*x)^(1/2)-9*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*c*d^4*e^2*((c*x+b)/b)^(1/2)*(-(
e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+4*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^2*d^
5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)
*b*e)^(1/2))*b^3*c*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+8*A*EllipticE(((c*x
+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^4*e^6*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1
/2)+8*A*x^4*b^2*c^2*e^6-9*B*x^2*c^4*d^5*e+54*A*x^3*c^4*d^3*e^3-9*B*x^3*c^4*d^4*e^2+8*A*EllipticE(((c*x+b)/b)^(
1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+2*B*
EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2
)*(-1/b*c*x)^(1/2)+3*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x
+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-3*B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^3*d^6*((c
*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+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+92*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x
+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-46*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b*c^3*d^4*
e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+8*A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*
d)*b*e)^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-24*A*EllipticF(
((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-
1/b*c*x)^(1/2)+5*B*x^2*b^3*c*d^2*e^4-41*A*x*b^2*c^2*d^3*e^3+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+15*A*x*b^3*c*d^2*e^4+4*B*EllipticE(((c*x+b)/b)^(1
/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/
2)+3*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e
-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c*d^2*e^4*((c*x
+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+2*B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(
1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-3*B*EllipticF(((c*x+
b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*
x)^(1/2)-23*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+
d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+4*A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c*d*e^
5*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+8*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)
*b*e)^(1/2))*x*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+6*B*EllipticE((
(c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c
*x)^(1/2)+2*B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)
/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+4*B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c^2*d^4*e^
2*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-6*B*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)
*b*e)^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-9*B*EllipticE(((c*x
+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c
*x)^(1/2)+16*A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d
)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-18*B*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c*d^3*e^
3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+46*A*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d
)*b*e)^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)-31*A*Ellipti
cE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(
-1/b*c*x)^(1/2)-12*A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2
)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)+8*A*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b
*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2))/(c*x+b)/x/(b*e-c*d)^3/(e*x+d)^(5
/2)/c/d^3/e

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((b*x + c*x^2)^(1/2)*(d + e*x)^(7/2)),x)

[Out]

int((A + B*x)/((b*x + c*x^2)^(1/2)*(d + e*x)^(7/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]

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]

Integral((A + B*x)/(sqrt(x*(b + c*x))*(d + e*x)**(7/2)), x)

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