3.1278 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=524 \[ -\frac {2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (b^2 c e (B d-A e)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+4 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (-b c (9 A e+4 B d)+8 A c^2 d+b^2 B e\right )+x \left (b^3 c e^2 (A e+4 B d)+b^2 c^2 d e (6 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-4 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]

[Out]

-2/3*(e*x+d)^(5/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)/b^2/c/(c*x^2+b*x)^(3/2)+2/3*(b*c*d^2*(8*A*c^2
*d+b^2*B*e-b*c*(9*A*e+4*B*d))+(16*A*c^4*d^3-4*b^4*B*e^3+b^3*c*e^2*(A*e+4*B*d)-8*b*c^3*d^2*(3*A*e+B*d)+b^2*c^2*
d*e*(6*A*e+5*B*d))*x)*(e*x+d)^(1/2)/b^4/c^2/(c*x^2+b*x)^(1/2)-2/3*(16*A*c^4*d^3-8*b^4*B*e^3+b^3*c*e^2*(2*A*e+5
*B*d)-8*b*c^3*d^2*(3*A*e+B*d)+b^2*c^2*d*e*(4*A*e+5*B*d))*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))
*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/c^(5/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*d*(-b*e+c*d)*(
16*A*c^3*d^2+4*b^3*B*e^2+b^2*c*e*(-A*e+B*d)-8*b*c^2*d*(2*A*e+B*d))*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c
/d)^(1/2))*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(7/2)/c^(5/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)

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Rubi [A]  time = 0.73, antiderivative size = 524, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {818, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {d+e x} \left (x \left (b^2 c^2 d e (6 A e+5 B d)+b^3 c e^2 (A e+4 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-4 b^4 B e^3\right )+b c d^2 \left (-b c (9 A e+4 B d)+8 A c^2 d+b^2 B e\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (b^2 c e (B d-A e)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+4 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 c^2 d e (4 A e+5 B d)+b^3 c e^2 (2 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^2*c*(b*x + c*x^2)^(3/2)) + (2*
Sqrt[d + e*x]*(b*c*d^2*(8*A*c^2*d + b^2*B*e - b*c*(4*B*d + 9*A*e)) + (16*A*c^4*d^3 - 4*b^4*B*e^3 + b^3*c*e^2*(
4*B*d + A*e) - 8*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 6*A*e))*x))/(3*b^4*c^2*Sqrt[b*x + c*x^2]) - (2
*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 4*
A*e))*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)^(7/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*d*(c*d - b*e)*(16*A*c^3*d^2 + 4*b^3*B*e^2 + b^2*c*
e*(B*d - A*e) - 8*b*c^2*d*(B*d + 2*A*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)])/(3*(-b)^(7/2)*c^(5/2)*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 818

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

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}}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d \left (4 b B c d-8 A c^2 d-b^2 B e+9 A b c e\right )+\frac {1}{2} e \left (2 A c^2 d+4 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c}\\ &=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b d e \left (8 A c^3 d^2-4 b^3 B e^2+b^2 c e (2 B d+A e)-b c^2 d (4 B d+11 A e)\right )-\frac {1}{4} e \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}\\ &=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}-\frac {\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}\\ &=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\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^4 c^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\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^4 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\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)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\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)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 4.77, size = 530, normalized size = 1.01 \[ -\frac {2 \left (b (d+e x) \left (x^2 (b+c x) (c d-b e)^2 \left (b c (5 B d-2 A e)-8 A c^2 d+5 b^2 B e\right )+c^2 d^2 x (b+c x)^2 (10 A b e-8 A c d+3 b B d)+b x^2 (b B-A c) (c d-b e)^3+A b c^2 d^3 (b+c x)^2\right )+x \sqrt {\frac {b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (c d-b e) \left (-b^2 c e (2 A e+B d)-b c^2 d (5 A e+4 B d)+8 A c^3 d^2+8 b^3 B e^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right )\right )\right )}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(b*(d + e*x)*(b*(b*B - A*c)*(c*d - b*e)^3*x^2 + (c*d - b*e)^2*(-8*A*c^2*d + 5*b^2*B*e + b*c*(5*B*d - 2*A*e
))*x^2*(b + c*x) + A*b*c^2*d^3*(b + c*x)^2 + c^2*d^2*(3*b*B*d - 8*A*c*d + 10*A*b*e)*x*(b + c*x)^2) + Sqrt[b/c]
*x*(b + c*x)*(Sqrt[b/c]*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d^2*(B*d + 3*A*e) +
b^2*c^2*d*e*(5*B*d + 4*A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*
e) - 8*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 4*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elli
pticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^3*d^2 + 8*b^3*B*e^2 - b^2*c*e*(B*d
+ 2*A*e) - b*c^2*d*(4*B*d + 5*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)])))/(3*b^5*c^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{3} x^{4} + A d^{3} + {\left (3 \, B d e^{2} + A e^{3}\right )} x^{3} + 3 \, {\left (B d^{2} e + A d e^{2}\right )} x^{2} + {\left (B d^{3} + 3 \, A d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral((B*e^3*x^4 + A*d^3 + (3*B*d*e^2 + A*e^3)*x^3 + 3*(B*d^2*e + A*d*e^2)*x^2 + (B*d^3 + 3*A*d^2*e)*x)*sqr
t(c*x^2 + b*x)*sqrt(e*x + d)/(c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2), x)

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maple [B]  time = 0.15, size = 3247, normalized size = 6.20 \[ \text {output too large to display} \]

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)^(5/2),x)

[Out]

2/3*(4*A*x^4*b^2*c^5*d*e^3+16*A*x^3*c^7*d^4-4*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2
)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^6*c*d*e^3+3*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)
*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^5*c^2*d^2*e^2+2*A*((c*x+b)
/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x
^2*b^4*c^3*d*e^3-28*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1
/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^4*d^2*e^2+40*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x
)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^2*c^5*d^3*e+15*A*((c*x+b)/b)^(1/2)*(-(e*x+d
)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^4*d^2*e^2
-32*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)
*b*e)^(1/2))*x^2*b^2*c^5*d^3*e+13*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(
((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^5*c^2*d*e^3-13*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/
2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^4*d^3*e-4*B*((c*x+b)/b)^(1/
2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^5*
c^2*d*e^3+3*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(
b*e-c*d)*b*e)^(1/2))*x^2*b^4*c^3*d^2*e^2+9*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*E
llipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^3*c^4*d^3*e+2*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d
)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^5*c^2*d*e^3+A*((c*x+b)/b)
^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*
b^4*c^3*d*e^3+9*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),
(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*c^3*d^3*e+A*x^3*b^4*c^3*e^4-8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*
(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c^4*d^4-4*B*x^3*b^5*c^2*e^4+16*A*x
^4*c^7*d^3*e+24*A*x^2*b*c^6*d^4-8*B*x^3*b*c^6*d^4-12*B*x^2*b^2*c^5*d^4+6*A*x*b^2*c^5*d^4-3*B*x*b^3*c^4*d^4-5*B
*x^4*b^4*c^3*e^4+2*A*x^4*b^3*c^4*e^4-28*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*Elli
pticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^4*c^3*d^2*e^2+40*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*
c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c^4*d^3*e+A*((c*x+b)/b)^(
1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^5*
c^2*d*e^3+15*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/
(b*e-c*d)*b*e)^(1/2))*x*b^4*c^3*d^2*e^2-32*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*E
llipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^3*c^4*d^3*e+13*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)
*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^6*c*d*e^3-13*B*((c*x+b)/b)
^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^
4*c^3*d^3*e+13*B*x^3*b^3*c^4*d^2*e^2-4*B*x^2*b^5*c^2*d*e^3-7*B*x^3*b^2*c^5*d^3*e+A*x^2*b^4*c^3*d*e^3-3*A*x^2*b
^3*c^4*d^2*e^2-31*A*x^2*b^2*c^5*d^3*e-8*B*x^4*b*c^6*d^3*e-24*A*x^4*b*c^6*d^2*e^2+2*B*x^2*b^4*c^3*d^2*e^2-11*A*
x*b^3*c^4*d^3*e+5*B*x^4*b^3*c^4*d*e^3+5*B*x^4*b^2*c^5*d^2*e^2+5*B*x^2*b^3*c^4*d^3*e+9*A*x^3*b^3*c^4*d*e^3-33*A
*x^3*b^2*c^5*d^2*e^2-3*B*x^3*b^4*c^3*d*e^3-A*b^3*c^4*d^4+2*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-
1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^5*c^2*e^4-16*A*((c*x+b)/b)^(1/2)*(-(
e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b*c^6*d^4+
16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*
b*e)^(1/2))*x^2*b*c^6*d^4-8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+
b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^6*c*e^4+8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*
x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^2*c^5*d^4-8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/
(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x^2*b^2*c^5*d^4+8*B*(
(c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(
1/2))*x*b^3*c^4*d^4+2*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^
(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^6*c*e^4-16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2
)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c^5*d^4+16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d
)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^2*c^5*d^4-8*B*((c*x+b)/b)
^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*x*b^
7*e^4)/x^2*((c*x+b)*x)^(1/2)/b^4/(c*x+b)^2/c^4/(e*x+d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(5/2),x)

[Out]

Timed out

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