3.15.79 \(\int \frac {(A+B x) (a+c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [1479]

Optimal. Leaf size=541 \[ \frac {4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac {8 \sqrt {-a} c^{3/2} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^2-12 A c d e+5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-2/15*(2*B*(a*d*e^2+4*c*d^3)-3*A*(-a*e^3+c*d^2*e)+e*(-6*A*c*d*e+5*B*a*e^2+11*B*c*d^2)*x)*(c*x^2+a)^(3/2)/e^2/(
a*e^2+c*d^2)/(e*x+d)^(5/2)+4/15*c*(32*B*c*d^3-12*A*c*d^2*e+29*a*B*d*e^2-9*a*A*e^3+e*(-3*A*c*d*e+5*B*a*e^2+8*B*
c*d^2)*x)*(c*x^2+a)^(1/2)/e^4/(a*e^2+c*d^2)/(e*x+d)^(1/2)+8/15*c^(3/2)*(-9*A*a*e^3-12*A*c*d^2*e+29*B*a*d*e^2+3
2*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)*(e*x+d)^(1/2)*(c*x^2/a+1)^(1/2)/e^5/(a*e^2+c*d^2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^
(1/2)))^(1/2)-8/15*(-12*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^(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/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.31, antiderivative size = 541, 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 = {825, 827, 858, 733, 435, 430} \begin {gather*} \frac {8 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-9 a A e^3+29 a B d e^2-12 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 )}{15 e^5 \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \left (5 a B e^2-12 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 )}{15 e^5 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}+\frac {4 c \sqrt {a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{15 e^4 \sqrt {d+e x} \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

(4*c*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3 + e*(8*B*c*d^2 - 3*A*c*d*e + 5*a*B*e^2)*x)*Sqrt[a +
c*x^2])/(15*e^4*(c*d^2 + a*e^2)*Sqrt[d + e*x]) - (2*(2*B*(4*c*d^3 + a*d*e^2) - 3*A*(c*d^2*e - a*e^3) + e*(11*
B*c*d^2 - 6*A*c*d*e + 5*a*B*e^2)*x)*(a + c*x^2)^(3/2))/(15*e^2*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) + (8*Sqrt[-a]*
c^(3/2)*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[Arc
Sin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^5*(c*d^2 + a*e^2)*Sqr
t[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*Sqrt[c]*(32*B*c*d^2 - 12*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)])/(15*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 825

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^
(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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 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)^{7/2}} \, dx &=-\frac {2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \int \frac {\left (3 a c e (B d-A e)-c \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^2 \left (c d^2+a e^2\right )}\\ &=\frac {4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac {4 \int \frac {a c e \left (8 B c d^2-3 A c d e+5 a B e^2\right )-c^2 \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e^4 \left (c d^2+a e^2\right )}\\ &=\frac {4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac {\left (4 c \left (32 B c d^2-12 A c d e+5 a B e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e^5}-\frac {\left (4 c^2 \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{15 e^5 \left (c d^2+a e^2\right )}\\ &=\frac {4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {\left (8 a c^{3/2} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 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 )}{15 \sqrt {-a} e^5 \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (8 a \sqrt {c} \left (32 B c d^2-12 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 )}{15 \sqrt {-a} e^5 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt {a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac {8 \sqrt {-a} c^{3/2} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 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 )}{15 e^5 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (32 B c d^2-12 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 )}{15 e^5 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 23.65, size = 705, normalized size = 1.30 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (5 B c+\frac {3 (B d-A e) \left (c d^2+a e^2\right )}{(d+e x)^3}+\frac {-17 B c d^2+12 A c d e-5 a B e^2}{(d+e x)^2}+\frac {c \left (73 B c d^3-33 A c d^2 e+61 a B d e^2-21 a A e^3\right )}{\left (c d^2+a e^2\right ) (d+e x)}\right )}{e^4}-\frac {8 c \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) \left (a+c x^2\right )+i \sqrt {c} \left (\sqrt {c} d+i \sqrt {a} e\right ) \left (-32 B c d^3+12 A c d^2 e-29 a B d e^2+9 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-12 A c d e+5 a B e^2+9 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}}} \left (c d^2+a e^2\right ) (d+e x)}\right )}{15 \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)^(7/2),x]

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(5*B*c + (3*(B*d - A*e)*(c*d^2 + a*e^2))/(d + e*x)^3 + (-17*B*c*d^2 + 12*A*c*d*
e - 5*a*B*e^2)/(d + e*x)^2 + (c*(73*B*c*d^3 - 33*A*c*d^2*e + 61*a*B*d*e^2 - 21*a*A*e^3))/((c*d^2 + a*e^2)*(d +
e*x))))/e^4 - (8*c*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^
3)*(a + c*x^2) + I*Sqrt[c]*(Sqrt[c]*d + I*Sqrt[a]*e)*(-32*B*c*d^3 + 12*A*c*d^2*e - 29*a*B*d*e^2 + 9*a*A*e^3)*S
qrt[(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 - 12*A*c*d*e + 5*
a*B*e^2 + (9*I)*Sqrt[a]*A*Sqrt[c]*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqr
t[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]]*(c*d^2 + a*e^2)*(d
+ e*x))))/(15*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(7382\) vs. \(2(469)=938\).
time = 0.86, size = 7383, normalized size = 13.65

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A a \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{7} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (12 A c d e -5 B \,e^{2} a -17 B c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{15 e^{6} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+a e \right ) c \left (21 A a \,e^{3}+33 A c \,d^{2} e -61 a B d \,e^{2}-73 B c \,d^{3}\right )}{15 e^{5} \left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 B c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{4}}+\frac {2 \left (-\frac {c \left (3 A c d e -2 B \,e^{2} a -6 B c \,d^{2}\right )}{e^{5}}+\frac {c \left (12 A c d e -5 B \,e^{2} a -17 B c \,d^{2}\right )}{15 e^{5}}+\frac {c^{2} d \left (21 A a \,e^{3}+33 A c \,d^{2} e -61 a B d \,e^{2}-73 B c \,d^{3}\right )}{15 e^{5} \left (e^{2} a +c \,d^{2}\right )}-\frac {a B c}{3 e^{3}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {c^{2} \left (A e -3 B d \right )}{e^{4}}+\frac {c^{2} \left (21 A a \,e^{3}+33 A c \,d^{2} e -61 a B d \,e^{2}-73 B c \,d^{3}\right )}{15 e^{4} \left (e^{2} a +c \,d^{2}\right )}-\frac {2 B \,c^{2} d}{3 e^{4}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(979\)
risch	\(\text {Expression too large to display}\)	\(2803\)
default	\(\text {Expression too large to display}\)	\(7383\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)/(x*e + d)^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.13, size = 835, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{7} + 15 \, B a^{2} x^{3} e^{7} - 9 \, {\left (2 \, A a c d x^{3} - 5 \, B a^{2} d x^{2}\right )} e^{6} + {\left (53 \, B a c d^{2} x^{3} - 54 \, A a c d^{2} x^{2} + 45 \, B a^{2} d^{2} x\right )} e^{5} - 3 \, {\left (4 \, A c^{2} d^{3} x^{3} - 53 \, B a c d^{3} x^{2} + 18 \, A a c d^{3} x - 5 \, B a^{2} d^{3}\right )} e^{4} + {\left (32 \, B c^{2} d^{4} x^{3} - 36 \, A c^{2} d^{4} x^{2} + 159 \, B a c d^{4} x - 18 \, A a c d^{4}\right )} e^{3} + {\left (96 \, B c^{2} d^{5} x^{2} - 36 \, A c^{2} d^{5} x + 53 \, B a c d^{5}\right )} e^{2} + 12 \, {\left (8 \, B c^{2} d^{6} x - A c^{2} d^{6}\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^{6} e - 9 \, A a c x^{3} e^{7} + {\left (29 \, B a c d x^{3} - 27 \, A a c d x^{2}\right )} e^{6} - 3 \, {\left (4 \, A c^{2} d^{2} x^{3} - 29 \, B a c d^{2} x^{2} + 9 \, A a c d^{2} x\right )} e^{5} + {\left (32 \, B c^{2} d^{3} x^{3} - 36 \, A c^{2} d^{3} x^{2} + 87 \, B a c d^{3} x - 9 \, A a c d^{3}\right )} e^{4} + {\left (96 \, B c^{2} d^{4} x^{2} - 36 \, A c^{2} d^{4} x + 29 \, B a c d^{4}\right )} e^{3} + 12 \, {\left (8 \, B c^{2} d^{5} x - A c^{2} d^{5}\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^{5} e^{2} + {\left (5 \, B a c x^{3} - 21 \, A a c x^{2} - 5 \, B a^{2} x - 3 \, A a^{2}\right )} e^{7} + 2 \, {\left (38 \, B a c d x^{2} - 15 \, A a c d x - B a^{2} d\right )} e^{6} + {\left (5 \, B c^{2} d^{2} x^{3} - 33 \, A c^{2} d^{2} x^{2} + 115 \, B a c d^{2} x - 15 \, A a c d^{2}\right )} e^{5} + 2 \, {\left (44 \, B c^{2} d^{3} x^{2} - 27 \, A c^{2} d^{3} x + 25 \, B a c d^{3}\right )} e^{4} + 24 \, {\left (6 \, B c^{2} d^{4} x - A c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{45 \, {\left (3 \, c d^{4} x e^{7} + c d^{5} e^{6} + a x^{3} e^{11} + 3 \, a d x^{2} e^{10} + {\left (c d^{2} x^{3} + 3 \, a d^{2} x\right )} e^{9} + {\left (3 \, c d^{3} x^{2} + a d^{3}\right )} e^{8}\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)^(7/2),x, algorithm="fricas")

[Out]

2/45*(4*(32*B*c^2*d^7 + 15*B*a^2*x^3*e^7 - 9*(2*A*a*c*d*x^3 - 5*B*a^2*d*x^2)*e^6 + (53*B*a*c*d^2*x^3 - 54*A*a*
c*d^2*x^2 + 45*B*a^2*d^2*x)*e^5 - 3*(4*A*c^2*d^3*x^3 - 53*B*a*c*d^3*x^2 + 18*A*a*c*d^3*x - 5*B*a^2*d^3)*e^4 +
(32*B*c^2*d^4*x^3 - 36*A*c^2*d^4*x^2 + 159*B*a*c*d^4*x - 18*A*a*c*d^4)*e^3 + (96*B*c^2*d^5*x^2 - 36*A*c^2*d^5*
x + 53*B*a*c*d^5)*e^2 + 12*(8*B*c^2*d^6*x - A*c^2*d^6)*e)*sqrt(c)*e^(1/2)*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)) + 12*(32*B*c^2*d^6*e - 9*A*a*c*x^3
*e^7 + (29*B*a*c*d*x^3 - 27*A*a*c*d*x^2)*e^6 - 3*(4*A*c^2*d^2*x^3 - 29*B*a*c*d^2*x^2 + 9*A*a*c*d^2*x)*e^5 + (3
2*B*c^2*d^3*x^3 - 36*A*c^2*d^3*x^2 + 87*B*a*c*d^3*x - 9*A*a*c*d^3)*e^4 + (96*B*c^2*d^4*x^2 - 36*A*c^2*d^4*x +
29*B*a*c*d^4)*e^3 + 12*(8*B*c^2*d^5*x - A*c^2*d^5)*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^5*e^2 + (5*B*a*c*x^3 - 21*A*a*c*x^2 - 5*B*a^2
*x - 3*A*a^2)*e^7 + 2*(38*B*a*c*d*x^2 - 15*A*a*c*d*x - B*a^2*d)*e^6 + (5*B*c^2*d^2*x^3 - 33*A*c^2*d^2*x^2 + 11
5*B*a*c*d^2*x - 15*A*a*c*d^2)*e^5 + 2*(44*B*c^2*d^3*x^2 - 27*A*c^2*d^3*x + 25*B*a*c*d^3)*e^4 + 24*(6*B*c^2*d^4
*x - A*c^2*d^4)*e^3)*sqrt(c*x^2 + a)*sqrt(x*e + d))/(3*c*d^4*x*e^7 + c*d^5*e^6 + a*x^3*e^11 + 3*a*d*x^2*e^10 +
(c*d^2*x^3 + 3*a*d^2*x)*e^9 + (3*c*d^3*x^2 + a*d^3)*e^8)

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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 {7}{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)**(7/2),x)

[Out]

Integral((A + B*x)*(a + c*x**2)**(3/2)/(d + e*x)**(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)/(x*e + d)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{7/2}} \,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)^(7/2),x)

[Out]

int(((a + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(7/2), x)

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