3.4.91 \(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx\) [391]

Optimal. Leaf size=398 \[ -\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {b x+c x^2}}{15 d e (c d-b e) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^2 e (c d-b e)^2 \sqrt {d+e x}}-\frac {4 \sqrt {-b} \sqrt {c} \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d e^2 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-4/15*(b^2*e^2-b*c*d*e+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^2/e^2/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/15*(-b*e+2*c*d)*Elli
pticF(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
/e^2/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/5*(c*x^2+b*x)^(1/2)/e/(e*x+d)^(5/2)+2/15*(-b*e+2*c*d)*(c*x^2
+b*x)^(1/2)/d/e/(-b*e+c*d)/(e*x+d)^(3/2)+4/15*(b^2*e^2-b*c*d*e+c^2*d^2)*(c*x^2+b*x)^(1/2)/d^2/e/(-b*e+c*d)^2/(
e*x+d)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 398, 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 = {746, 848, 857, 729, 113, 111, 118, 117} \begin {gather*} -\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^2 \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} (2 c d-b e) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d e^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{15 d^2 e \sqrt {d+e x} (c d-b e)^2}+\frac {2 \sqrt {b x+c x^2} (2 c d-b e)}{15 d e (d+e x)^{3/2} (c d-b e)}-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[b*x + c*x^2])/(5*e*(d + e*x)^(5/2)) + (2*(2*c*d - b*e)*Sqrt[b*x + c*x^2])/(15*d*e*(c*d - b*e)*(d + e*
x)^(3/2)) + (4*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[b*x + c*x^2])/(15*d^2*e*(c*d - b*e)^2*Sqrt[d + e*x]) - (4*Sq
rt[-b]*Sqrt[c]*(c^2*d^2 - b*c*d*e + 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^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b
]*Sqrt[c]*(2*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*e^2*(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 746

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 5.88, size = 362, normalized size = 0.91 \begin {gather*} -\frac {2 \left (b e x (b+c x) \left (-b^2 e^3 x (5 d+2 e x)-c^2 d^2 \left (d^2+6 d e x+2 e^2 x^2\right )+b c d e \left (-d^2+7 d e x+2 e^2 x^2\right )\right )+\sqrt {\frac {b}{c}} c (d+e x)^2 \left (2 \sqrt {\frac {b}{c}} \left (c^2 d^2-b c d e+b^2 e^2\right ) (b+c x) (d+e x)+2 i b e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} 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^2 e^2 (c d-b e)^2 \sqrt {x (b+c x)} (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*e*x*(b + c*x)*(-(b^2*e^3*x*(5*d + 2*e*x)) - c^2*d^2*(d^2 + 6*d*e*x + 2*e^2*x^2) + b*c*d*e*(-d^2 + 7*d*e
*x + 2*e^2*x^2)) + Sqrt[b/c]*c*(d + e*x)^2*(2*Sqrt[b/c]*(c^2*d^2 - b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + (2
*I)*b*e*(c^2*d^2 - b*c*d*e + 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^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*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^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*
x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1896\) vs. \(2(338)=676\).
time = 0.45, size = 1897, normalized size = 4.77

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{5 e^{4} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{15 e^{3} d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {4 \left (c e \,x^{2}+b e x \right ) \left (b^{2} e^{2}-b c d e +d^{2} c^{2}\right )}{15 d^{2} \left (b e -c d \right )^{2} e^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c \left (b e -2 c d \right )}{15 e^{2} d \left (b e -c d \right )}+\frac {-\frac {2}{15} b c d e +\frac {2}{15} d^{2} c^{2}+\frac {2}{15} b^{2} e^{2}}{e^{2} \left (b e -c d \right ) d^{2}}-\frac {2 b \left (b^{2} e^{2}-b c d e +d^{2} c^{2}\right )}{15 e \,d^{2} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}-\frac {4 \left (b^{2} e^{2}-b c d e +d^{2} c^{2}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{15 e \,d^{2} \left (b e -c d \right )^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(641\)
default	\(\text {Expression too large to display}\)	\(1897\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(-6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b^2*c^2*d^3*e^2*x+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+
b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^4*e*x-8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/
2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^2*e^3*x+8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^3*e^2*x-4*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^4*e*
x+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*b^4*d*e^4*x+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e+((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3*e^2-3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/
b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e-4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3*e^2+6*c^4*d^3*e^2*x^3
+c^4*d^4*e*x^2+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b^2*c^2*d^2*e^3*x^2-2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^3*e^2*x^2+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^2*e^3*x+((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d*e^4*x^2-3*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^
2*c^2*d^2*e^3*x^2+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),
(b*e/(b*e-c*d))^(1/2))*b*c^3*d^3*e^2*x^2-4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d*e^4*x^2+2*b^2*c^2*e^5*x^4+2*c^4*d^2*e^3*x^4+2*b^3*c*e^5*
x^3-2*b*c^3*d*e^4*x^4+3*b^2*c^2*d*e^4*x^3-5*b*c^3*d^2*e^3*x^3+5*b^3*c*d*e^4*x^2-7*b^2*c^2*d^2*e^3*x^2+7*b*c^3*
d^3*e^2*x^2+b^2*c^2*d^3*e^2*x+b*c^3*d^4*e*x+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*e^5*x^2+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*
(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^5+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b
*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*d^2*e^3-2*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^5)*(
x*(c*x+b))^(1/2)/c/(b*e-c*d)^2/d^2/(c*x+b)/x/e^2/(e*x+d)^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.35, size = 845, normalized size = 2.12 \begin {gather*} \frac {2 \, {\left ({\left (2 \, c^{3} d^{6} + 2 \, b^{3} x^{3} e^{6} - 3 \, {\left (b^{2} c d x^{3} - 2 \, b^{3} d x^{2}\right )} e^{5} - 3 \, {\left (b c^{2} d^{2} x^{3} + 3 \, b^{2} c d^{2} x^{2} - 2 \, b^{3} d^{2} x\right )} e^{4} + {\left (2 \, c^{3} d^{3} x^{3} - 9 \, b c^{2} d^{3} x^{2} - 9 \, b^{2} c d^{3} x + 2 \, b^{3} d^{3}\right )} e^{3} + 3 \, {\left (2 \, c^{3} d^{4} x^{2} - 3 \, b c^{2} d^{4} x - b^{2} c d^{4}\right )} e^{2} + 3 \, {\left (2 \, c^{3} d^{5} x - b c^{2} d^{5}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 6 \, {\left (c^{3} d^{5} e + b^{2} c x^{3} e^{6} - {\left (b c^{2} d x^{3} - 3 \, b^{2} c d x^{2}\right )} e^{5} + {\left (c^{3} d^{2} x^{3} - 3 \, b c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x\right )} e^{4} + {\left (3 \, c^{3} d^{3} x^{2} - 3 \, b c^{2} d^{3} x + b^{2} c d^{3}\right )} e^{3} + {\left (3 \, c^{3} d^{4} x - b c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (c^{3} d^{4} e^{2} + 2 \, b^{2} c x^{2} e^{6} - {\left (2 \, b c^{2} d x^{2} - 5 \, b^{2} c d x\right )} e^{5} + {\left (2 \, c^{3} d^{2} x^{2} - 7 \, b c^{2} d^{2} x\right )} e^{4} + {\left (6 \, c^{3} d^{3} x + b c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{45 \, {\left (c^{3} d^{7} e^{3} + b^{2} c d^{2} x^{3} e^{8} - {\left (2 \, b c^{2} d^{3} x^{3} - 3 \, b^{2} c d^{3} x^{2}\right )} e^{7} + {\left (c^{3} d^{4} x^{3} - 6 \, b c^{2} d^{4} x^{2} + 3 \, b^{2} c d^{4} x\right )} e^{6} + {\left (3 \, c^{3} d^{5} x^{2} - 6 \, b c^{2} d^{5} x + b^{2} c d^{5}\right )} e^{5} + {\left (3 \, c^{3} d^{6} x - 2 \, b c^{2} d^{6}\right )} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*((2*c^3*d^6 + 2*b^3*x^3*e^6 - 3*(b^2*c*d*x^3 - 2*b^3*d*x^2)*e^5 - 3*(b*c^2*d^2*x^3 + 3*b^2*c*d^2*x^2 - 2*
b^3*d^2*x)*e^4 + (2*c^3*d^3*x^3 - 9*b*c^2*d^3*x^2 - 9*b^2*c*d^3*x + 2*b^3*d^3)*e^3 + 3*(2*c^3*d^4*x^2 - 3*b*c^
2*d^4*x - b^2*c*d^4)*e^2 + 3*(2*c^3*d^5*x - b*c^2*d^5)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b
*c*d*e + b^2*e^2)*e^(-2)/c^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)*e^(-3)/c^3, 1/3*(c
*d + (3*c*x + b)*e)*e^(-1)/c) + 6*(c^3*d^5*e + b^2*c*x^3*e^6 - (b*c^2*d*x^3 - 3*b^2*c*d*x^2)*e^5 + (c^3*d^2*x^
3 - 3*b*c^2*d^2*x^2 + 3*b^2*c*d^2*x)*e^4 + (3*c^3*d^3*x^2 - 3*b*c^2*d^3*x + b^2*c*d^3)*e^3 + (3*c^3*d^4*x - b*
c^2*d^4)*e^2)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^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)*e^(-3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*
e^(-2)/c^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)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e
)*e^(-1)/c)) + 3*(c^3*d^4*e^2 + 2*b^2*c*x^2*e^6 - (2*b*c^2*d*x^2 - 5*b^2*c*d*x)*e^5 + (2*c^3*d^2*x^2 - 7*b*c^2
*d^2*x)*e^4 + (6*c^3*d^3*x + b*c^2*d^3)*e^3)*sqrt(c*x^2 + b*x)*sqrt(x*e + d))/(c^3*d^7*e^3 + b^2*c*d^2*x^3*e^8
- (2*b*c^2*d^3*x^3 - 3*b^2*c*d^3*x^2)*e^7 + (c^3*d^4*x^3 - 6*b*c^2*d^4*x^2 + 3*b^2*c*d^4*x)*e^6 + (3*c^3*d^5*
x^2 - 6*b*c^2*d^5*x + b^2*c*d^5)*e^5 + (3*c^3*d^6*x - 2*b*c^2*d^6)*e^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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