\(\int \frac {(b x+c x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [402]

Optimal result

Integrand size = 23, antiderivative size = 401 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(c*x^2+b*x)^(5/2)/e/(e*x+d)^(3/2)+10/21*(2*c*e*x-7*b*e+16*c*d)*(c*x^2+b*x)^(3/2)/e^3/(e*x+d)^(1/2)-2/21*(
-b*e+2*c*d)*(3*b^2*e^2-128*b*c*d*e+128*c^2*d^2)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/
2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/e^6/c^(1/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+4/21*d*(-b*e+c*d)*(27*b
^2*e^2-128*b*c*d*e+128*c^2*d^2)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*
x/b)^(1/2)*(1+e*x/d)^(1/2)/e^6/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/21*(128*c^2*d^2-176*b*c*d*e+51*b^2*e^
2-48*c*e*(-b*e+2*c*d)*x)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {746, 826, 828, 857, 729, 113, 111, 118, 117} \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (51 b^2 e^2-48 c e x (2 c d-b e)-176 b c d e+128 c^2 d^2\right )}{21 e^5}+\frac {10 \left (b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]

[In]

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

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 - 176*b*c*d*e + 51*b^2*e^2 - 48*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(21*e^5)
+ (10*(16*c*d - 7*b*e + 2*c*e*x)*(b*x + c*x^2)^(3/2))/(21*e^3*Sqrt[d + e*x]) - (2*(b*x + c*x^2)^(5/2))/(3*e*(
d + e*x)^(3/2)) - (2*Sqrt[-b]*(2*c*d - b*e)*(128*c^2*d^2 - 128*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)])/(21*Sqrt[c]*e^6*Sqrt[1 + (e*x)/d]*Sq
rt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e + 27*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b
]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(21*Sqrt[c]*e^6*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 826

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

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (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 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e} \\ & = \frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {10 \int \frac {\left (\frac {1}{2} b (16 c d-7 b e)+8 c (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^3} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {4 \int \frac {-\frac {1}{4} b c d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 c e^5} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{21 e^6}+\frac {\left (2 d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 e^6} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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}{21 e^6 \sqrt {b x+c x^2}}+\frac {\left (2 d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{21 e^6 \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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}{21 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (2 d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\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}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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 )}{21 \sqrt {c} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\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 )}{21 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.33 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.10 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (x (b+c x))^{5/2} \left (-\frac {\left (256 c^3 d^3-384 b c^2 d^2 e+134 b^2 c d e^2-3 b^3 e^3\right ) (b+c x) (d+e x)}{c \sqrt {x}}+\frac {e \sqrt {x} (b+c x) \left (b^2 e^2 \left (51 d^2+67 d e x+9 e^2 x^2\right )+b c e \left (-176 d^3-224 d^2 e x-25 d e^2 x^2+9 e^3 x^3\right )+c^2 \left (128 d^4+160 d^3 e x+16 d^2 e^2 x^2-6 d e^3 x^3+3 e^4 x^4\right )\right )}{d+e x}+i \sqrt {\frac {b}{c}} e \left (-256 c^3 d^3+384 b c^2 d^2 e-134 b^2 c d e^2+3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \sqrt {\frac {b}{c}} e \left (128 c^3 d^3-208 b c^2 d^2 e+83 b^2 c d e^2-3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{21 e^6 x^{5/2} (b+c x)^3 \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(x*(b + c*x))^(5/2)*(-(((256*c^3*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*(b + c*x)*(d + e*x))/
(c*Sqrt[x])) + (e*Sqrt[x]*(b + c*x)*(b^2*e^2*(51*d^2 + 67*d*e*x + 9*e^2*x^2) + b*c*e*(-176*d^3 - 224*d^2*e*x -
25*d*e^2*x^2 + 9*e^3*x^3) + c^2*(128*d^4 + 160*d^3*e*x + 16*d^2*e^2*x^2 - 6*d*e^3*x^3 + 3*e^4*x^4)))/(d + e*x
) + I*Sqrt[b/c]*e*(-256*c^3*d^3 + 384*b*c^2*d^2*e - 134*b^2*c*d*e^2 + 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/
(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*e*(128*c^3*d^3 - 208*b*c^2*d^2*e +
83*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*
d)/(b*e)]))/(21*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1134\) vs. \(2(341)=682\).

Time = 2.35 (sec) , antiderivative size = 1135, normalized size of antiderivative = 2.83

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1135\)
default	\(\text {Expression too large to display}\)	\(1692\)

[In]

int((c*x^2+b*x)^(5/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(x*(e*x+d)*(c*x+b))^(1/2)/x/(c*x+b)*(-2/3*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/e^
7*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^2+14/3*(c*e*x^2+b*e*x)*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^6/((x
+d/e)*(c*e*x^2+b*e*x))^(1/2)+2/7*c^2/e^3*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/5*(c^2/e^3*(3*b*e-2*c*d)-
2/7*c^2/e^3*(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(3*c/e^4*(b^2*e^2-2*b*c*d*e+c^2*d^2
)-5/7*c^2/e^3*b*d-2/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^
2+c*d*x^2+b*d*x)^(1/2)+2*(-d*(2*b^3*e^3-9*b^2*c*d*e^2+12*b*c^2*d^2*e-5*c^3*d^3)/e^6-1/3*d^2*(b^2*e^2-2*b*c*d*e
+c^2*d^2)/e^6*c+7/3*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^6*(b*e-c*d)-7/3*b/e^5*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)-1/
3*(3*c/e^4*(b^2*e^2-2*b*c*d*e+c^2*d^2)-5/7*c^2/e^3*b*d-2/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/e^3*(3*b*e+3*c*d))/c
/e*(2*b*e+2*c*d))/c/e*b*d)/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+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(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))+2*(1/e^5*(b^3*e^3-6*b^2*
c*d*e^2+9*b*c^2*d^2*e-4*c^3*d^3)-7/3*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^5*c-3/5*(c^2/e^3*(3*b*e-2*c*d)-2/7*c^2/
e^3*(3*b*e+3*c*d))/c/e*b*d-2/3*(3*c/e^4*(b^2*e^2-2*b*c*d*e+c^2*d^2)-5/7*c^2/e^3*b*d-2/5*(c^2/e^3*(3*b*e-2*c*d)
-2/7*c^2/e^3*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(b*e+c*d))/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+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)*((-1/c*b+d/e)*EllipticE(((1/c*b+x)*c/b)^(1/2),(-1/
c*b/(-1/c*b+d/e))^(1/2))-d/e*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.17 (sec) , antiderivative size = 806, normalized size of antiderivative = 2.01 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left ({\left (256 \, c^{4} d^{6} - 512 \, b c^{3} d^{5} e + 278 \, b^{2} c^{2} d^{4} e^{2} - 22 \, b^{3} c d^{3} e^{3} - 3 \, b^{4} d^{2} e^{4} + {\left (256 \, c^{4} d^{4} e^{2} - 512 \, b c^{3} d^{3} e^{3} + 278 \, b^{2} c^{2} d^{2} e^{4} - 22 \, b^{3} c d e^{5} - 3 \, b^{4} e^{6}\right )} x^{2} + 2 \, {\left (256 \, c^{4} d^{5} e - 512 \, b c^{3} d^{4} e^{2} + 278 \, b^{2} c^{2} d^{3} e^{3} - 22 \, b^{3} c d^{2} e^{4} - 3 \, b^{4} d e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{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 )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (256 \, c^{4} d^{5} e - 384 \, b c^{3} d^{4} e^{2} + 134 \, b^{2} c^{2} d^{3} e^{3} - 3 \, b^{3} c d^{2} e^{4} + {\left (256 \, c^{4} d^{3} e^{3} - 384 \, b c^{3} d^{2} e^{4} + 134 \, b^{2} c^{2} d e^{5} - 3 \, b^{3} c e^{6}\right )} x^{2} + 2 \, {\left (256 \, c^{4} d^{4} e^{2} - 384 \, b c^{3} d^{3} e^{3} + 134 \, b^{2} c^{2} d^{2} e^{4} - 3 \, b^{3} c d e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{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 )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{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 )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (3 \, c^{4} e^{6} x^{4} + 128 \, c^{4} d^{4} e^{2} - 176 \, b c^{3} d^{3} e^{3} + 51 \, b^{2} c^{2} d^{2} e^{4} - 3 \, {\left (2 \, c^{4} d e^{5} - 3 \, b c^{3} e^{6}\right )} x^{3} + {\left (16 \, c^{4} d^{2} e^{4} - 25 \, b c^{3} d e^{5} + 9 \, b^{2} c^{2} e^{6}\right )} x^{2} + {\left (160 \, c^{4} d^{3} e^{3} - 224 \, b c^{3} d^{2} e^{4} + 67 \, b^{2} c^{2} d e^{5}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{63 \, {\left (c^{2} e^{9} x^{2} + 2 \, c^{2} d e^{8} x + c^{2} d^{2} e^{7}\right )}} \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/63*((256*c^4*d^6 - 512*b*c^3*d^5*e + 278*b^2*c^2*d^4*e^2 - 22*b^3*c*d^3*e^3 - 3*b^4*d^2*e^4 + (256*c^4*d^4*e
^2 - 512*b*c^3*d^3*e^3 + 278*b^2*c^2*d^2*e^4 - 22*b^3*c*d*e^5 - 3*b^4*e^6)*x^2 + 2*(256*c^4*d^5*e - 512*b*c^3*
d^4*e^2 + 278*b^2*c^2*d^3*e^3 - 22*b^3*c*d^2*e^4 - 3*b^4*d*e^5)*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*(256*c^4*d^5*e - 384*b*c^3*d^4*e^2 + 134*b^2*c^2*d^3*e^3 - 3*b^3*c*d^2*e^4 + (
256*c^4*d^3*e^3 - 384*b*c^3*d^2*e^4 + 134*b^2*c^2*d*e^5 - 3*b^3*c*e^6)*x^2 + 2*(256*c^4*d^4*e^2 - 384*b*c^3*d^
3*e^3 + 134*b^2*c^2*d^2*e^4 - 3*b^3*c*d*e^5)*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*(3*c^4*e^6*x^4 + 128*c^4*d^4*e^2 - 176*b*c^3*d^3*e^3 + 51*b^2*c^2*d^2*e^
4 - 3*(2*c^4*d*e^5 - 3*b*c^3*e^6)*x^3 + (16*c^4*d^2*e^4 - 25*b*c^3*d*e^5 + 9*b^2*c^2*e^6)*x^2 + (160*c^4*d^3*e
^3 - 224*b*c^3*d^2*e^4 + 67*b^2*c^2*d*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^2*e^9*x^2 + 2*c^2*d*e^8*x +
c^2*d^2*e^7)

Sympy [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)/(d + e*x)**(5/2), x)

Maxima [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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