3.5.1 \(\int \frac {(b x+c x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\) [401]

Optimal. Leaf size=457 \[ -\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \sqrt {-b} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2*(c*x^2+b*x)^(5/2)/e/(e*x+d)^(1/2)-10/63*(-14*c*e*x-15*b*e+16*c*d)*(c*x^2+b*x)^(3/2)*(e*x+d)^(1/2)/e^3+4/63*
(-b^4*e^4-7*b^3*c*d*e^3+135*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),
(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/c^(3/2)/e^6/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2
)-2/63*d*(-b*e+c*d)*(-b*e+2*c*d)*(-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)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1/2)/c^(3/2)/e^6/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/6
3*(128*c^3*d^3-240*b*c^2*d^2*e+111*b^2*c*d*e^2-b^3*e^3-3*c*e*(b^2*e^2-32*b*c*d*e+32*c^2*d^2)*x)*(e*x+d)^(1/2)*
(c*x^2+b*x)^(1/2)/c/e^5

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Rubi [A]
time = 0.39, antiderivative size = 457, 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, 828, 857, 729, 113, 111, 118, 117} \begin {gather*} -\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} 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 (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{63 c e^5}-\frac {10 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3 - 3*c*e*(32*c^2*d^2 - 32*b*c*d*e
+ b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(63*c*e^5) - (10*Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*x)*(b*x + c*x^2)^(3/
2))/(63*e^3) - (2*(b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (4*Sqrt[-b]*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^
2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqr
t[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e
)*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/2)*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 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*} \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{e}\\ &=-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {10 \int \frac {\left (-\frac {1}{2} b c d (16 c d-15 b e)-\frac {1}{2} c \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{21 c e^3}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} b c d \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+\frac {1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 c^2 e^5}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 c e^6}+\frac {\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{63 c e^6}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-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}{63 c e^6 \sqrt {b x+c x^2}}+\frac {\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{63 c e^6 \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\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}{63 c e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-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}{63 c e^6 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \sqrt {-b} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\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 )}{63 c^{3/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-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 )}{63 c^{3/2} e^6 \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 = 21.57, size = 498, normalized size = 1.09 \begin {gather*} \frac {2 (x (b+c x))^{5/2} \left (\frac {2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) (b+c x) (d+e x)}{c \sqrt {x}}-e \sqrt {x} (b+c x) \left (-b^3 e^3 (d+e x)+3 b^2 c e^2 \left (37 d^2+11 d e x-5 e^2 x^2\right )-b c^2 e \left (240 d^3+64 d^2 e x-31 d e^2 x^2+19 e^3 x^3\right )+c^3 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )-2 i \sqrt {\frac {b}{c}} e \left (-128 c^4 d^4+256 b c^3 d^3 e-135 b^2 c^2 d^2 e^2+7 b^3 c d e^3+b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \sqrt {\frac {b}{c}} e \left (-128 c^4 d^4+272 b c^3 d^3 e-159 b^2 c^2 d^2 e^2+13 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )}{63 c e^6 x^{5/2} (b+c x)^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*((2*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*(b
+ c*x)*(d + e*x))/(c*Sqrt[x]) - e*Sqrt[x]*(b + c*x)*(-(b^3*e^3*(d + e*x)) + 3*b^2*c*e^2*(37*d^2 + 11*d*e*x - 5
*e^2*x^2) - b*c^2*e*(240*d^3 + 64*d^2*e*x - 31*d*e^2*x^2 + 19*e^3*x^3) + c^3*(128*d^4 + 32*d^3*e*x - 16*d^2*e^
2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)) - (2*I)*Sqrt[b/c]*e*(-128*c^4*d^4 + 256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 +
7*b^3*c*d*e^3 + b^4*e^4)*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^4*d^4 + 272*b*c^3*d^3*e - 159*b^2*c^2*d^2*e^2 + 13*b^3*c*d*e^3 + 2*b^4*e^4)*Sqr
t[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(63*c*e^6*x^(5/2)*(b
+ c*x)^3*Sqrt[d + e*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1169\) vs. \(2(399)=798\).
time = 0.53, size = 1170, normalized size = 2.56

method	result	size
default	\(\frac {2 \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}\, \left (240 b^{2} c^{4} d^{3} e^{2} x -111 b^{3} c^{3} d^{2} e^{3} x -128 b \,c^{5} d^{4} e x -64 b^{2} c^{4} d \,e^{4} x^{3}+80 b \,c^{5} d^{2} e^{3} x^{3}-32 b^{3} c^{3} d \,e^{4} x^{2}-47 b^{2} c^{4} d^{2} e^{3} x^{2}+208 b \,c^{5} d^{3} e^{2} x^{2}+b^{4} c^{2} d \,e^{4} x +782 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c^{3} d^{3} e^{2}-768 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{4} d^{4} e -510 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c^{3} d^{3} e^{2}+640 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{4} d^{4} e -284 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{4} c^{2} d^{2} e^{3}+\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{5} c d \,e^{4}+125 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{4} c^{2} d^{2} e^{3}-41 b \,c^{5} d \,e^{4} x^{4}+12 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{5} c d \,e^{4}-32 c^{6} d^{3} e^{2} x^{3}+b^{4} c^{2} e^{5} x^{2}-128 c^{6} d^{4} e \,x^{2}+7 c^{6} e^{5} x^{6}-10 c^{6} d \,e^{4} x^{5}+34 b^{2} c^{4} e^{5} x^{4}+16 c^{6} d^{2} e^{3} x^{4}+16 b^{3} c^{3} e^{5} x^{3}+26 b \,c^{5} e^{5} x^{5}+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{6} e^{5}-256 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{5} d^{5}+256 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{5} d^{5}\right )}{63 c^{3} e^{6} x \left (c e \,x^{2}+b e x +c d x +b d \right )}\)	\(1170\)
elliptic	\(\text {Expression too large to display}\)	\(1532\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(5/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/63*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(240*b^2*c^4*d^3*e^2*x-111*b^3*c^3*d^2*e^3*x+782*((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^3*d^3*e^2-768*(
(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^4*d^4*e-128*b*c^5*d^4*e*x-64*b^2*c^4*d*e^4*x^3+80*b*c^5*d^2*e^3*x^3-32*b^3*c^3*d*e^4*x^2-47*b^2*c^4*d^
2*e^3*x^2+208*b*c^5*d^3*e^2*x^2+12*((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^5*c*d*e^4+b^4*c^2*d*e^4*x-510*((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^3*d^3*e^2+640*((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^4*d^4*e-
284*((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*c^2*d^2*e^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^5*c*d*e^4+125*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^2*d^2*e^3-41*b*c^5*d*e^4*x^4-32*c^6*d^3*e^2*x^3+b^4*c^2*e^5
*x^2-128*c^6*d^4*e*x^2+7*c^6*e^5*x^6+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^6*e^5-10*c^6*d*e^4*x^5+34*b^2*c^4*e^5*x^4+16*c^6*d^2*e^3*x^4+16*b^
3*c^3*e^5*x^3+26*b*c^5*e^5*x^5-256*((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^5*d^5+256*((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^5*d^5)/c^3/e^6/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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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)^(5/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(x*e + d)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.42, size = 718, normalized size = 1.57 \begin {gather*} -\frac {2 \, {\left ({\left (256 \, c^{5} d^{6} - 2 \, b^{5} x e^{6} - {\left (13 \, b^{4} c d x + 2 \, b^{5} d\right )} e^{5} - {\left (77 \, b^{3} c^{2} d^{2} x + 13 \, b^{4} c d^{2}\right )} e^{4} + {\left (478 \, b^{2} c^{3} d^{3} x - 77 \, b^{3} c^{2} d^{3}\right )} e^{3} - 2 \, {\left (320 \, b c^{4} d^{4} x - 239 \, b^{2} c^{3} d^{4}\right )} e^{2} + 128 \, {\left (2 \, c^{5} d^{5} x - 5 \, b c^{4} 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 (128 \, c^{5} d^{5} e - b^{4} c x e^{6} - {\left (7 \, b^{3} c^{2} d x + b^{4} c d\right )} e^{5} + {\left (135 \, b^{2} c^{3} d^{2} x - 7 \, b^{3} c^{2} d^{2}\right )} e^{4} - {\left (256 \, b c^{4} d^{3} x - 135 \, b^{2} c^{3} d^{3}\right )} e^{3} + 128 \, {\left (c^{5} d^{4} x - 2 \, b c^{4} 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 (128 \, c^{5} d^{4} e^{2} - {\left (7 \, c^{5} x^{4} + 19 \, b c^{4} x^{3} + 15 \, b^{2} c^{3} x^{2} + b^{3} c^{2} x\right )} e^{6} + {\left (10 \, c^{5} d x^{3} + 31 \, b c^{4} d x^{2} + 33 \, b^{2} c^{3} d x - b^{3} c^{2} d\right )} e^{5} - {\left (16 \, c^{5} d^{2} x^{2} + 64 \, b c^{4} d^{2} x - 111 \, b^{2} c^{3} d^{2}\right )} e^{4} + 16 \, {\left (2 \, c^{5} d^{3} x - 15 \, b c^{4} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{189 \, {\left (c^{3} x e^{8} + c^{3} d e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-2/189*((256*c^5*d^6 - 2*b^5*x*e^6 - (13*b^4*c*d*x + 2*b^5*d)*e^5 - (77*b^3*c^2*d^2*x + 13*b^4*c*d^2)*e^4 + (4
78*b^2*c^3*d^3*x - 77*b^3*c^2*d^3)*e^3 - 2*(320*b*c^4*d^4*x - 239*b^2*c^3*d^4)*e^2 + 128*(2*c^5*d^5*x - 5*b*c^
4*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*(128*c^5*d^5*e
- b^4*c*x*e^6 - (7*b^3*c^2*d*x + b^4*c*d)*e^5 + (135*b^2*c^3*d^2*x - 7*b^3*c^2*d^2)*e^4 - (256*b*c^4*d^3*x -
135*b^2*c^3*d^3)*e^3 + 128*(c^5*d^4*x - 2*b*c^4*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, weierstras
sPInverse(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*(128*c^5*d^4*e^2 - (7*c^5*x^4 + 19*b*c^4*x^3 + 15
*b^2*c^3*x^2 + b^3*c^2*x)*e^6 + (10*c^5*d*x^3 + 31*b*c^4*d*x^2 + 33*b^2*c^3*d*x - b^3*c^2*d)*e^5 - (16*c^5*d^2
*x^2 + 64*b*c^4*d^2*x - 111*b^2*c^3*d^2)*e^4 + 16*(2*c^5*d^3*x - 15*b*c^4*d^3)*e^3)*sqrt(c*x^2 + b*x)*sqrt(x*e
+ d))/(c^3*x*e^8 + c^3*d*e^7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)/(d + e*x)**(3/2), 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)^(5/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(x*e + d)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x)

[Out]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(3/2), x)

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