3.4.99 \(\int \sqrt {d+e x} (b x+c x^2)^{5/2} \, dx\) [399]

Optimal. Leaf size=666 \[ \frac {2 \sqrt {d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{9009 c^3 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac {4 \sqrt {-b} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

2/13*(e*x+d)^(3/2)*(c*x^2+b*x)^(5/2)/e+10/9009*(16*c^3*d^3-31*b*c^2*d^2*e+9*b^2*c*d*e^2-18*b^3*e^3-14*c*e*(3*b
^2*e^2-b*c*d*e+c^2*d^2)*x)*(c*x^2+b*x)^(3/2)*(e*x+d)^(1/2)/c^2/e^3-10/143*(-b*e+2*c*d)*(c*x^2+b*x)^(5/2)*(e*x+
d)^(1/2)/c/e-4/9009*(24*b^6*e^6-20*b^5*c*d*e^5-21*b^4*c^2*d^2*e^4-46*b^3*c^3*d^3*e^3+343*b^2*c^4*d^4*e^2-384*b
*c^5*d^5*e+128*c^6*d^6)*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^(7/2)/e^6/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/9009*d*(-b*e+c*d)*(-b*e+2*c*d)*(24*b^4*e^4+49
*b^3*c*d*e^3+79*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)*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^(7/2)/e^6/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/9009*(12
8*c^5*d^5-368*b*c^4*d^4*e+303*b^2*c^3*d^3*e^2-22*b^3*c^2*d^2*e^3-17*b^4*c*d*e^4+24*b^5*e^5-3*c*e*(-24*b^4*e^4+
11*b^3*c*d*e^3+21*b^2*c^2*d^2*e^2-64*b*c^3*d^3*e+32*c^4*d^4)*x)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c^3/e^5

________________________________________________________________________________________

Rubi [A]
time = 0.57, antiderivative size = 666, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {748, 846, 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 (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{9009 c^{7/2} 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 (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {10 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{9009 c^2 e^3}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{9009 c^3 e^5}+\frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {10 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{143 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(128*c^5*d^5 - 368*b*c^4*d^4*e + 303*b^2*c^3*d^3*e^2 - 22*b^3*c^2*d^2*e^3 - 17*b^4*c*d*e^4 +
24*b^5*e^5 - 3*c*e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 21*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 24*b^4*e^4)*x)*Sqrt[b*
x + c*x^2])/(9009*c^3*e^5) + (10*Sqrt[d + e*x]*(16*c^3*d^3 - 31*b*c^2*d^2*e + 9*b^2*c*d*e^2 - 18*b^3*e^3 - 14*
c*e*(c^2*d^2 - b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/(9009*c^2*e^3) - (10*(2*c*d - b*e)*Sqrt[d + e*x]*(
b*x + c*x^2)^(5/2))/(143*c*e) + (2*(d + e*x)^(3/2)*(b*x + c*x^2)^(5/2))/(13*e) - (4*Sqrt[-b]*(128*c^6*d^6 - 38
4*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*S
qrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/
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^4*d^4 - 256*b*c^3
*d^3*e + 79*b^2*c^2*d^2*e^2 + 49*b^3*c*d*e^3 + 24*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellipti
cF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/2)*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 748

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 + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, 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] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[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 846

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

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 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx &=\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac {5 \int \sqrt {d+e x} (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2} \, dx}{13 e}\\ &=-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac {10 \int \frac {\left (\frac {1}{2} b d (c d+5 b e)+\left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{143 c e}\\ &=\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}+\frac {20 \int \frac {\left (-\frac {1}{4} b d \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3\right )-\frac {1}{4} \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{3003 c^2 e^3}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{9009 c^3 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac {8 \int \frac {\frac {1}{8} b d \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5\right )+\frac {1}{4} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{9009 c^3 e^5}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{9009 c^3 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}+\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{9009 c^3 e^6}-\frac {\left (2 \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{9009 c^3 e^6}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{9009 c^3 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}+\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{9009 c^3 e^6 \sqrt {b x+c x^2}}-\frac {\left (2 \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{9009 c^3 e^6 \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{9009 c^3 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac {\left (2 \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\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}{9009 c^3 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^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4\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}{9009 c^3 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{9009 c^3 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac {4 \sqrt {-b} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\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 )}{9009 c^{7/2} 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^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4\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 )}{9009 c^{7/2} 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 = 18.17, size = 663, normalized size = 1.00 \begin {gather*} \frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (24 b^5 e^5-b^4 c e^4 (17 d+18 e x)+b^3 c^2 e^3 \left (-22 d^2+12 d e x+15 e^2 x^2\right )+b^2 c^3 e^2 \left (303 d^3-218 d^2 e x+178 d e^2 x^2+1113 e^3 x^3\right )+b c^4 e \left (-368 d^4+272 d^3 e x-225 d^2 e^2 x^2+196 d e^3 x^3+1701 e^4 x^4\right )+c^5 \left (128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right )\right )+\sqrt {\frac {b}{c}} \left (-2 \sqrt {\frac {b}{c}} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) (b+c x) (d+e x)-2 i b e \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\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 (128 c^6 d^6-400 b c^5 d^5 e+383 b^2 c^4 d^4 e^2-70 b^3 c^3 d^3 e^3-25 b^4 c^2 d^2 e^4-64 b^5 c d e^5+48 b^6 e^6\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 )}{9009 b c^3 e^6 x^3 (b+c x)^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(24*b^5*e^5 - b^4*c*e^4*(17*d + 18*e*x) + b^3*c^2*e^3*(-22*d
^2 + 12*d*e*x + 15*e^2*x^2) + b^2*c^3*e^2*(303*d^3 - 218*d^2*e*x + 178*d*e^2*x^2 + 1113*e^3*x^3) + b*c^4*e*(-3
68*d^4 + 272*d^3*e*x - 225*d^2*e^2*x^2 + 196*d*e^3*x^3 + 1701*e^4*x^4) + c^5*(128*d^5 - 96*d^4*e*x + 80*d^3*e^
2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)) + Sqrt[b/c]*(-2*Sqrt[b/c]*(128*c^6*d^6 - 384*b*c^5*d^5*e
+ 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*(b + c*x)*(d +
e*x) - (2*I)*b*e*(128*c^6*d^6 - 384*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e
^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sq
rt[x]], (c*d)/(b*e)] + I*b*e*(128*c^6*d^6 - 400*b*c^5*d^5*e + 383*b^2*c^4*d^4*e^2 - 70*b^3*c^3*d^3*e^3 - 25*b^
4*c^2*d^2*e^4 - 64*b^5*c*d*e^5 + 48*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[S
qrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(9009*b*c^3*e^6*x^3*(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. \(1727\) vs. \(2(600)=1200\).
time = 0.44, size = 1728, normalized size = 2.59

method	result	size
default	\(\text {Expression too large to display}\)	\(1728\)
elliptic	\(\text {Expression too large to display}\)	\(2334\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(5/2)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9009*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-50*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^3*d^3*e^4+778*((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^4*d^4*e^3-1454*((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^5*d^5*e^2-
23*((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^6*c^2*d^2*e^5-20*((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^3*d^3*e^4-395*((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^4*c^4*d^4*e^3+1054*((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^5*d^5*e^2-896*((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^6*d^6*
e-88*((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^7*c*d*e^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*c^2*d^2*e^5+24*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^7*c*d*e^6+1024*((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^6*d^6*e-36*b*c^7*d^2*e^5*x^5+1318*b^3
*c^5*d*e^6*x^4-69*b^2*c^6*d^2*e^5*x^4+57*b*c^7*d^3*e^4*x^4-8*b^4*c^4*d*e^6*x^3-50*b^3*c^5*d^2*e^5*x^3+132*b^2*
c^6*d^3*e^4*x^3-112*b*c^7*d^4*e^3*x^3-11*b^5*c^3*d*e^6*x^2-27*b^4*c^4*d^2*e^5*x^2+63*b^3*c^5*d^3*e^4*x^2+207*b
^2*c^6*d^4*e^3*x^2-336*b*c^7*d^5*e^2*x^2+24*b^6*c^2*d*e^6*x+2653*b*c^7*d*e^6*x^6+3188*b^2*c^6*d*e^6*x^5+48*((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^8*e^7+2394*b*c^7*e^7*x^7+756*c^8*d*e^6*x^7+2814*b^2*c^6*e^7*x^6-7*c^8*d^2*e^5*x^6+1128*b^3*c^5*e^7*x^5+10*c^
8*d^3*e^4*x^5-3*b^4*c^4*e^7*x^4-16*c^8*d^4*e^3*x^4+6*b^5*c^3*e^7*x^3+32*c^8*d^5*e^2*x^3+24*b^6*c^2*e^7*x^2+693
*c^8*e^7*x^8+128*c^8*d^6*e*x^2-17*b^5*c^3*d^2*e^5*x-22*b^4*c^4*d^3*e^4*x+303*b^3*c^5*d^4*e^3*x-368*b^2*c^6*d^5
*e^2*x+128*b*c^7*d^6*e*x+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^7*d^7-256*((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*c^7*d^7)/c^5/x/(c*e*x^2+b*e*x+c*d*x+b*d)/e^6

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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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.87, size = 702, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left ({\left (256 \, c^{7} d^{7} - 896 \, b c^{6} d^{6} e + 1022 \, b^{2} c^{5} d^{5} e^{2} - 315 \, b^{3} c^{4} d^{4} e^{3} - 68 \, b^{4} c^{3} d^{3} e^{4} - 31 \, b^{5} c^{2} d^{2} e^{5} - 64 \, b^{6} c d e^{6} + 48 \, b^{7} e^{7}\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^{7} d^{6} e - 384 \, b c^{6} d^{5} e^{2} + 343 \, b^{2} c^{5} d^{4} e^{3} - 46 \, b^{3} c^{4} d^{3} e^{4} - 21 \, b^{4} c^{3} d^{2} e^{5} - 20 \, b^{5} c^{2} d e^{6} + 24 \, b^{6} c e^{7}\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^{7} d^{5} e^{2} + 3 \, {\left (231 \, c^{7} x^{5} + 567 \, b c^{6} x^{4} + 371 \, b^{2} c^{5} x^{3} + 5 \, b^{3} c^{4} x^{2} - 6 \, b^{4} c^{3} x + 8 \, b^{5} c^{2}\right )} e^{7} + {\left (63 \, c^{7} d x^{4} + 196 \, b c^{6} d x^{3} + 178 \, b^{2} c^{5} d x^{2} + 12 \, b^{3} c^{4} d x - 17 \, b^{4} c^{3} d\right )} e^{6} - {\left (70 \, c^{7} d^{2} x^{3} + 225 \, b c^{6} d^{2} x^{2} + 218 \, b^{2} c^{5} d^{2} x + 22 \, b^{3} c^{4} d^{2}\right )} e^{5} + {\left (80 \, c^{7} d^{3} x^{2} + 272 \, b c^{6} d^{3} x + 303 \, b^{2} c^{5} d^{3}\right )} e^{4} - 16 \, {\left (6 \, c^{7} d^{4} x + 23 \, b c^{6} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-7\right )}}{27027 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/27027*((256*c^7*d^7 - 896*b*c^6*d^6*e + 1022*b^2*c^5*d^5*e^2 - 315*b^3*c^4*d^4*e^3 - 68*b^4*c^3*d^3*e^4 - 31
*b^5*c^2*d^2*e^5 - 64*b^6*c*d*e^6 + 48*b^7*e^7)*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^7*d^6*e - 384*b*c^6*d^5*e^2 + 343*b^2*c^5*d^4*e^3 - 46*b^3*c^4*d^3*e^4 - 21*b^4
*c^3*d^2*e^5 - 20*b^5*c^2*d*e^6 + 24*b^6*c*e^7)*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*(128*c^7*d^5*e^2 + 3*(231*c^7*x^5 + 567*b*c^6*x^4 + 371*b^2
*c^5*x^3 + 5*b^3*c^4*x^2 - 6*b^4*c^3*x + 8*b^5*c^2)*e^7 + (63*c^7*d*x^4 + 196*b*c^6*d*x^3 + 178*b^2*c^5*d*x^2
+ 12*b^3*c^4*d*x - 17*b^4*c^3*d)*e^6 - (70*c^7*d^2*x^3 + 225*b*c^6*d^2*x^2 + 218*b^2*c^5*d^2*x + 22*b^3*c^4*d^
2)*e^5 + (80*c^7*d^3*x^2 + 272*b*c^6*d^3*x + 303*b^2*c^5*d^3)*e^4 - 16*(6*c^7*d^4*x + 23*b*c^6*d^4)*e^3)*sqrt(
c*x^2 + b*x)*sqrt(x*e + d))*e^(-7)/c^5

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}\, 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)**(1/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*sqrt(d + e*x), 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)^(1/2),x, algorithm="giac")

[Out]

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

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

[Out]

int((b*x + c*x^2)^(5/2)*(d + e*x)^(1/2), x)

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