\(\int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx\) [1259]

Optimal result

Integrand size = 28, antiderivative size = 494 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {2 \sqrt {-b} \sqrt {c} \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^3 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (B d (8 c d-9 b e)+A e (2 c d-b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 d e^3 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/15*(2*A*e*(b^2*e^2-b*c*d*e+c^2*d^2)+B*d*(3*b^2*e^2-13*b*c*d*e+8*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)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/d^2/e^3/(-b*e+c*d)^2/(1+e*x/d)^(
1/2)/(c*x^2+b*x)^(1/2)+2/15*(B*d*(-9*b*e+8*c*d)+A*e*(-b*e+2*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)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/d/e^3/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x
)^(1/2)-2/15*(d*(B*d*(-3*b*e+4*c*d)+A*e*(-2*b*e+c*d))+e*(B*d*(-6*b*e+7*c*d)-A*e*(-b*e+2*c*d))*x)*(c*x^2+b*x)^(
1/2)/d/e^2/(-b*e+c*d)/(e*x+d)^(5/2)+2/15*(2*A*e*(b^2*e^2-b*c*d*e+c^2*d^2)+B*d*(3*b^2*e^2-13*b*c*d*e+8*c^2*d^2)
)*(c*x^2+b*x)^(1/2)/d^2/e^2/(-b*e+c*d)^2/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {824, 848, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (A e (2 c d-b e)+B d (8 c d-9 b e)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 d e^3 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right )}{15 d^2 e^2 \sqrt {d+e x} (c d-b e)^2}-\frac {2 \sqrt {b x+c x^2} (e x (B d (7 c d-6 b e)-A e (2 c d-b e))+d (A e (c d-2 b e)+B d (4 c d-3 b e)))}{15 d e^2 (d+e x)^{5/2} (c d-b e)} \]

[In]

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

[Out]

(2*(2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*e^2))*Sqrt[b*x + c*x^2])/(15*d^2
*e^2*(c*d - b*e)^2*Sqrt[d + e*x]) - (2*(d*(B*d*(4*c*d - 3*b*e) + A*e*(c*d - 2*b*e)) + e*(B*d*(7*c*d - 6*b*e) -
A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(15*d*e^2*(c*d - b*e)*(d + e*x)^(5/2)) - (2*Sqrt[-b]*Sqrt[c]*(2*A*e*
(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*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^2*e^3*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sq
rt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[c]*(B*d*(8*c*d - 9*b*e) + A*e*(2*c*d - b*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d*e^3*(c*d - b*e)*Sqrt[d + e*x]
*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], 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 113

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 117

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

Rule 118

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 729

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], 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] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 848

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 857

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*} \text {integral}& = -\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {1}{2} b \left (2 b e \left (\frac {3 B d}{2}+A e\right )-c d (4 B d+A e)\right )-\frac {1}{2} c (2 c d (4 B d+A e)-b e (9 B d+A e)) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{15 d e^2 (c d-b e)} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac {4 \int \frac {-\frac {1}{4} b c d \left (A e (c d+b e)+B \left (4 c d^2-6 b d e\right )\right )-\frac {1}{4} c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 d^2 e^2 (c d-b e)^2} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac {(c (B d (8 c d-9 b e)+A e (2 c d-b e))) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 d e^3 (c d-b e)}-\frac {\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 d^2 e^3 (c d-b e)^2} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac {\left (c (B d (8 c d-9 b e)+A e (2 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 e^3 (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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^2 e^3 (c d-b e)^2 \sqrt {b x+c x^2}} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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^2 e^3 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c (B d (8 c d-9 b e)+A e (2 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 e^3 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {2 \sqrt {-b} \sqrt {c} \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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^2 e^3 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (B d (8 c d-9 b e)+A e (2 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 e^3 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 19.60 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (b e x (b+c x) \left (3 d^2 (B d-A e) (c d-b e)^2-d (c d-b e) (B d (7 c d-6 b e)+A e (-2 c d+b e)) (d+e x)+\left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) (d+e x)^2\right )-\sqrt {\frac {b}{c}} c (d+e x)^2 \left (\sqrt {\frac {b}{c}} \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) (b+c x) (d+e x)+i b e \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) (B d (4 c d-3 b e)+A e (c d-2 b e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{15 b d^2 e^3 (c d-b e)^2 \sqrt {x (b+c x)} (d+e x)^{5/2}} \]

[In]

Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^(7/2),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)*(B*d*(7*c*d - 6*b*e) + A*e*(-2*c*d + b*e)
)*(d + e*x) + (2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*e^2))*(d + e*x)^2) -
Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*b^2*
e^2))*(b + c*x)*(d + e*x) + I*b*e*(2*A*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*d*(8*c^2*d^2 - 13*b*c*d*e + 3*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*b*e*
(c*d - b*e)*(B*d*(4*c*d - 3*b*e) + A*e*(c*d - 2*b*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^2*e^3*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))

Maple [A] (verified)

Time = 2.26 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.69

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (A e -B d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{5} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (A b \,e^{2}-2 A c d e -6 B b d e +7 B c \,d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{15 e^{4} d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}\right )}{15 d^{2} \left (b e -c d \right )^{2} e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {B c}{e^{3}}+\frac {c \left (A b \,e^{2}-2 A c d e -6 B b d e +7 B c \,d^{2}\right )}{15 e^{3} d \left (b e -c d \right )}+\frac {2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}}{15 e^{3} \left (b e -c d \right ) d^{2}}-\frac {b \left (2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}\right )}{15 e^{2} d^{2} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}-\frac {2 \left (2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{15 e^{2} d^{2} \left (b e -c d \right )^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(835\)
default	\(\text {Expression too large to display}\)	\(3831\)

[In]

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

[Out]

1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((e*x+d)*x*(c*x+b))^(1/2)/x/(c*x+b)*(-2/5*(A*e-B*d)/e^5*(c*e*x^3+b*e*x^2+c*d
*x^2+b*d*x)^(1/2)/(x+d/e)^3+2/15*(A*b*e^2-2*A*c*d*e-6*B*b*d*e+7*B*c*d^2)/e^4/d/(b*e-c*d)*(c*e*x^3+b*e*x^2+c*d*
x^2+b*d*x)^(1/2)/(x+d/e)^2+2/15*(c*e*x^2+b*e*x)/d^2/(b*e-c*d)^2/e^3*(2*A*b^2*e^3-2*A*b*c*d*e^2+2*A*c^2*d^2*e+3
*B*b^2*d*e^2-13*B*b*c*d^2*e+8*B*c^2*d^3)/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2*(B*c/e^3+1/15*c*(A*b*e^2-2*A*c*d*e-
6*B*b*d*e+7*B*c*d^2)/e^3/d/(b*e-c*d)+1/15/e^3/(b*e-c*d)*(2*A*b^2*e^3-2*A*b*c*d*e^2+2*A*c^2*d^2*e+3*B*b^2*d*e^2
-13*B*b*c*d^2*e+8*B*c^2*d^3)/d^2-1/15*b/e^2/d^2/(b*e-c*d)^2*(2*A*b^2*e^3-2*A*b*c*d*e^2+2*A*c^2*d^2*e+3*B*b^2*d
*e^2-13*B*b*c*d^2*e+8*B*c^2*d^3))*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b
*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))-2/15/e^2*(2*A*b^2*e^3-2*A*b
*c*d*e^2+2*A*c^2*d^2*e+3*B*b^2*d*e^2-13*B*b*c*d^2*e+8*B*c^2*d^3)/d^2/(b*e-c*d)^2*b*((x+b/c)/b*c)^(1/2)*((x+d/e
)/(-b/c+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-b/c+d/e)*EllipticE(((x+b/c)/b*c)^(
1/2),(-b/c/(-b/c+d/e))^(1/2))-d/e*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.20 (sec) , antiderivative size = 1269, normalized size of antiderivative = 2.57 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/45*((8*B*c^3*d^7 + 2*A*b^3*d^3*e^4 - (17*B*b*c^2 - 2*A*c^3)*d^6*e + (8*B*b^2*c - 3*A*b*c^2)*d^5*e^2 + 3*(B*b
^3 - A*b^2*c)*d^4*e^3 + (8*B*c^3*d^4*e^3 + 2*A*b^3*e^7 - (17*B*b*c^2 - 2*A*c^3)*d^3*e^4 + (8*B*b^2*c - 3*A*b*c
^2)*d^2*e^5 + 3*(B*b^3 - A*b^2*c)*d*e^6)*x^3 + 3*(8*B*c^3*d^5*e^2 + 2*A*b^3*d*e^6 - (17*B*b*c^2 - 2*A*c^3)*d^4
*e^3 + (8*B*b^2*c - 3*A*b*c^2)*d^3*e^4 + 3*(B*b^3 - A*b^2*c)*d^2*e^5)*x^2 + 3*(8*B*c^3*d^6*e + 2*A*b^3*d^2*e^5
- (17*B*b*c^2 - 2*A*c^3)*d^5*e^2 + (8*B*b^2*c - 3*A*b*c^2)*d^4*e^3 + 3*(B*b^3 - A*b^2*c)*d^3*e^4)*x)*sqrt(c*e
)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*
d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(8*B*c^3*d^6*e + 2*A*b^2*c*d^3*e^4 - (13*B*
b*c^2 - 2*A*c^3)*d^5*e^2 + (3*B*b^2*c - 2*A*b*c^2)*d^4*e^3 + (8*B*c^3*d^3*e^4 + 2*A*b^2*c*e^7 - (13*B*b*c^2 -
2*A*c^3)*d^2*e^5 + (3*B*b^2*c - 2*A*b*c^2)*d*e^6)*x^3 + 3*(8*B*c^3*d^4*e^3 + 2*A*b^2*c*d*e^6 - (13*B*b*c^2 - 2
*A*c^3)*d^3*e^4 + (3*B*b^2*c - 2*A*b*c^2)*d^2*e^5)*x^2 + 3*(8*B*c^3*d^5*e^2 + 2*A*b^2*c*d^2*e^5 - (13*B*b*c^2
- 2*A*c^3)*d^4*e^3 + (3*B*b^2*c - 2*A*b*c^2)*d^3*e^4)*x)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^
2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse
(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c
^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(4*B*c^3*d^5*e^2 + A*b*c^2*d^3*e^4 - (6*B*b*c^2 - A*c^3)*d^4*e^
3 + (8*B*c^3*d^3*e^4 + 2*A*b^2*c*e^7 - (13*B*b*c^2 - 2*A*c^3)*d^2*e^5 + (3*B*b^2*c - 2*A*b*c^2)*d*e^6)*x^2 + (
9*B*c^3*d^4*e^3 - 7*A*b*c^2*d^2*e^5 + 5*A*b^2*c*d*e^6 - (13*B*b*c^2 - 6*A*c^3)*d^3*e^4)*x)*sqrt(c*x^2 + b*x)*s
qrt(e*x + d))/(c^3*d^7*e^4 - 2*b*c^2*d^6*e^5 + b^2*c*d^5*e^6 + (c^3*d^4*e^7 - 2*b*c^2*d^3*e^8 + b^2*c*d^2*e^9)
*x^3 + 3*(c^3*d^5*e^6 - 2*b*c^2*d^4*e^7 + b^2*c*d^3*e^8)*x^2 + 3*(c^3*d^6*e^5 - 2*b*c^2*d^5*e^6 + b^2*c*d^4*e^
7)*x)

Sympy [F]

\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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