3.15.77 \(\int \frac {(A+B x) (a+c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [1477]

Optimal. Leaf size=448 \[ \frac {4 \sqrt {d+e x} \left (5 a B e^2+4 c d (8 B d-7 A e)-3 c e (8 B d-7 A e) x\right ) \sqrt {a+c x^2}}{35 e^4}+\frac {2 (8 B d-7 A e+B e x) \left (a+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^3-28 A c d^2 e+29 a B d e^2-21 a A e^3\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 B c d^2-28 A c d e+5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

2/7*(B*e*x-7*A*e+8*B*d)*(c*x^2+a)^(3/2)/e^2/(e*x+d)^(1/2)+4/35*(5*a*B*e^2+4*c*d*(-7*A*e+8*B*d)-3*c*e*(-7*A*e+8
*B*d)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^4+8/35*(-21*A*a*e^3-28*A*c*d^2*e+29*B*a*d*e^2+32*B*c*d^3)*EllipticE(1
/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(e*x+
d)^(1/2)*(c*x^2/a+1)^(1/2)/e^5/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-8/35*(a*e^2+c*
d^2)*(-28*A*c*d*e+5*B*a*e^2+32*B*c*d^2)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(
-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(c*x^2/a+1)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/e^5/c
^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {827, 829, 858, 733, 435, 430} \begin {gather*} -\frac {8 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \left (5 a B e^2-28 A c d e+32 B c d^2\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 \sqrt {c} e^5 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {8 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-21 a A e^3+29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^5 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}}+\frac {4 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{35 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(4*Sqrt[d + e*x]*(5*a*B*e^2 + 4*c*d*(8*B*d - 7*A*e) - 3*c*e*(8*B*d - 7*A*e)*x)*Sqrt[a + c*x^2])/(35*e^4) + (2*
(8*B*d - 7*A*e + B*e*x)*(a + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) + (8*Sqrt[-a]*Sqrt[c]*(32*B*c*d^3 - 28*A*c*d^
2*e + 29*a*B*d*e^2 - 21*a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[
-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(35*e^5*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)
]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2 + a*e^2)*(32*B*c*d^2 - 28*A*c*d*e + 5*a*B*e^2)*Sqrt[(Sqrt[c]*(d + e*x)
)/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*
a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(35*Sqrt[c]*e^5*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*a*Rt[-c/a, 2]*(d + e*x)^m*(Sqrt[1
+ c*(x^2/a)]/(c*Sqrt[a + c*x^2]*(c*((d + e*x)/(c*d - a*e*Rt[-c/a, 2])))^m)), Subst[Int[(1 + 2*a*e*Rt[-c/a, 2]*
(x^2/(c*d - a*e*Rt[-c/a, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]], x] /; FreeQ[{a, c, d, e},
x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + 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 829

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

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=\frac {2 (8 B d-7 A e+B e x) \left (a+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {6 \int \frac {(-a B e+c (8 B d-7 A e) x) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^2}\\ &=\frac {4 \sqrt {d+e x} \left (5 a B e^2+4 c d (8 B d-7 A e)-3 c e (8 B d-7 A e) x\right ) \sqrt {a+c x^2}}{35 e^4}+\frac {2 (8 B d-7 A e+B e x) \left (a+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {1}{2} a c e \left (8 B c d^2-7 A c d e+5 a B e^2\right )+\frac {1}{2} c^2 \left (32 B c d^3-28 A c d^2 e+29 a B d e^2-21 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{35 c e^4}\\ &=\frac {4 \sqrt {d+e x} \left (5 a B e^2+4 c d (8 B d-7 A e)-3 c e (8 B d-7 A e) x\right ) \sqrt {a+c x^2}}{35 e^4}+\frac {2 (8 B d-7 A e+B e x) \left (a+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {\left (4 \left (c d^2+a e^2\right ) \left (32 B c d^2-28 A c d e+5 a B e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{35 e^5}-\frac {\left (4 c \left (32 B c d^3-28 A c d^2 e+29 a B d e^2-21 a A e^3\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{35 e^5}\\ &=\frac {4 \sqrt {d+e x} \left (5 a B e^2+4 c d (8 B d-7 A e)-3 c e (8 B d-7 A e) x\right ) \sqrt {a+c x^2}}{35 e^4}+\frac {2 (8 B d-7 A e+B e x) \left (a+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {\left (8 a \sqrt {c} \left (32 B c d^3-28 A c d^2 e+29 a B d e^2-21 a A e^3\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{35 \sqrt {-a} e^5 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (8 a \left (c d^2+a e^2\right ) \left (32 B c d^2-28 A c d e+5 a B e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{35 \sqrt {-a} \sqrt {c} e^5 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {4 \sqrt {d+e x} \left (5 a B e^2+4 c d (8 B d-7 A e)-3 c e (8 B d-7 A e) x\right ) \sqrt {a+c x^2}}{35 e^4}+\frac {2 (8 B d-7 A e+B e x) \left (a+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^3-28 A c d^2 e+29 a B d e^2-21 a A e^3\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 B c d^2-28 A c d e+5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 23.34, size = 661, normalized size = 1.48 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (-7 A e \left (5 a e^2+c \left (8 d^2+2 d e x-e^2 x^2\right )\right )+B \left (5 a e^2 (10 d+3 e x)+c \left (64 d^3+16 d^2 e x-8 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{e^4 (d+e x)}+\frac {8 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-32 B c d^3+28 A c d^2 e-29 a B d e^2+21 a A e^3\right ) \left (a+c x^2\right )+\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \left (-32 B c d^3+28 A c d^2 e-29 a B d e^2+21 a A e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (32 B c d^2-24 i \sqrt {a} B \sqrt {c} d e-28 A c d e+5 a B e^2+21 i \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^6 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{35 \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(-7*A*e*(5*a*e^2 + c*(8*d^2 + 2*d*e*x - e^2*x^2)) + B*(5*a*e^2*(10*d + 3*e*x) +
c*(64*d^3 + 16*d^2*e*x - 8*d*e^2*x^2 + 5*e^3*x^3))))/(e^4*(d + e*x)) + (8*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c
]]*(-32*B*c*d^3 + 28*A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*(a + c*x^2) + Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)
*(-32*B*c*d^3 + 28*A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-
(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]
]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(
32*B*c*d^2 - (24*I)*Sqrt[a]*B*Sqrt[c]*d*e - 28*A*c*d*e + 5*a*B*e^2 + (21*I)*Sqrt[a]*A*Sqrt[c]*e^2)*Sqrt[(e*((I
*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[
I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)
]))/(e^6*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(35*Sqrt[a + c*x^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2560\) vs. \(2(376)=752\).
time = 0.81, size = 2561, normalized size = 5.72 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/35*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(-14*A*c^3*d*e^4*x^3-8*B*c^3*d*e^4*x^4-128*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*
d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)
*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^
(1/2)*c^2*d^4*e-196*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/
2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*
c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2*d^2*e^3+116*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*
x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e
*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d*e^4+244*B*(-(e
*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/(
(-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1
/2)*e+c*d))^(1/2))*a*c^2*d^3*e^2-56*A*a*c^2*d^2*e^3+112*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c
/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*c^2*d^3*e^2-148*
B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2
))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a
*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a*c*d^2*e^3+128*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^
(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^3*d^5+16*B*c^3*d^2*e^3*x^3-28
*A*a*c^2*e^5*x^2+84*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/
2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*
c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*e^5-84*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a
*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)
*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*e^5-112*A*(-(e*x+d)*c
/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^
(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c
*d))^(1/2))*c^3*d^4*e-20*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d)
)^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-
((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a^2*e^5-35*A*a^2*c*e^5+20*B*a*c^2*e^5*x^3+64*B*
a*c^2*d^3*e^2-96*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*
((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^
(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2*d^3*e^2+64*B*c^3*d^3*e^2*x^2-56*A*c^3*d^2*e^3*x^2+5*B*c^3*e^5*
x^5+7*A*c^3*e^5*x^4-96*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^
(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((
-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d*e^4+84*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*
x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e
*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2*d^2*e^3+112*A*(-
(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e
/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^
(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a*c*d*e^4+42*B*a*c^2*d*e^4*x^2-14*A*a*c^2*d*e^4*x+16*B*a*c^2*d^2*e^3*x+50*B*
a^2*c*d*e^4+15*B*a^2*c*e^5*x)/c/e^6/(c*e*x^3+c*d*x^2+a*e*x+a*d)

________________________________________________________________________________________

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((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)/(x*e + d)^(3/2), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.72, size = 473, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{5} + 15 \, B a^{2} x e^{5} - 3 \, {\left (14 \, A a c d x - 5 \, B a^{2} d\right )} e^{4} + {\left (53 \, B a c d^{2} x - 42 \, A a c d^{2}\right )} e^{3} - {\left (28 \, A c^{2} d^{3} x - 53 \, B a c d^{3}\right )} e^{2} + 4 \, {\left (8 \, B c^{2} d^{4} x - 7 \, A c^{2} d^{4}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 12 \, {\left (32 \, B c^{2} d^{4} e - 21 \, A a c x e^{5} + {\left (29 \, B a c d x - 21 \, A a c d\right )} e^{4} - {\left (28 \, A c^{2} d^{2} x - 29 \, B a c d^{2}\right )} e^{3} + 4 \, {\left (8 \, B c^{2} d^{3} x - 7 \, A c^{2} d^{3}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (64 \, B c^{2} d^{3} e^{2} + {\left (5 \, B c^{2} x^{3} + 7 \, A c^{2} x^{2} + 15 \, B a c x - 35 \, A a c\right )} e^{5} - 2 \, {\left (4 \, B c^{2} d x^{2} + 7 \, A c^{2} d x - 25 \, B a c d\right )} e^{4} + 8 \, {\left (2 \, B c^{2} d^{2} x - 7 \, A c^{2} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{105 \, {\left (c x e^{7} + c d e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/105*(4*(32*B*c^2*d^5 + 15*B*a^2*x*e^5 - 3*(14*A*a*c*d*x - 5*B*a^2*d)*e^4 + (53*B*a*c*d^2*x - 42*A*a*c*d^2)*e
^3 - (28*A*c^2*d^3*x - 53*B*a*c*d^3)*e^2 + 4*(8*B*c^2*d^4*x - 7*A*c^2*d^4)*e)*sqrt(c)*e^(1/2)*weierstrassPInve
rse(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*e^2)*e^(-3)/c, 1/3*(3*x*e + d)*e^(-1)) + 12*(32*B*c^2
*d^4*e - 21*A*a*c*x*e^5 + (29*B*a*c*d*x - 21*A*a*c*d)*e^4 - (28*A*c^2*d^2*x - 29*B*a*c*d^2)*e^3 + 4*(8*B*c^2*d
^3*x - 7*A*c^2*d^3)*e^2)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*
e^2)*e^(-3)/c, weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*e^2)*e^(-3)/c, 1/3*(3*
x*e + d)*e^(-1))) + 3*(64*B*c^2*d^3*e^2 + (5*B*c^2*x^3 + 7*A*c^2*x^2 + 15*B*a*c*x - 35*A*a*c)*e^5 - 2*(4*B*c^2
*d*x^2 + 7*A*c^2*d*x - 25*B*a*c*d)*e^4 + 8*(2*B*c^2*d^2*x - 7*A*c^2*d^2)*e^3)*sqrt(c*x^2 + a)*sqrt(x*e + d))/(
c*x*e^7 + c*d*e^6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)*(a + c*x**2)**(3/2)/(d + e*x)**(3/2), x)

________________________________________________________________________________________

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((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)/(x*e + d)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(3/2),x)

[Out]

int(((a + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(3/2), x)

________________________________________________________________________________________