3.1637 \(\int \frac{(b+2 c x) (d+e x)^{5/2}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=600 \[ -\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{21 c^3 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{21 c^2}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{7 c} \]

[Out]

(4*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(21*c^2) + (2*(2*c*d - b
*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(7*c) + (4*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/7 + (Sqrt[2]*Sqrt
[b^2 - 4*a*c]*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2
*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(21*c^3*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^
2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 + 2*b
^2*e^2 - c*e*(3*b*d + 5*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x
^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sq
rt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(21*c^3*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.892539, antiderivative size = 600, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {832, 843, 718, 424, 419} \[ \frac{4 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{21 c^2}-\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(21*c^2) + (2*(2*c*d - b
*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(7*c) + (4*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/7 + (Sqrt[2]*Sqrt
[b^2 - 4*a*c]*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2
*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(21*c^3*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^
2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 + 2*b
^2*e^2 - c*e*(3*b*d + 5*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x
^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sq
rt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(21*c^3*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 832

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 843

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]

Rule 718

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

Rule 424

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

Rule 419

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

Rubi steps

\begin{align*} \int \frac{(b+2 c x) (d+e x)^{5/2}}{\sqrt{a+b x+c x^2}} \, dx &=\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{2 \int \frac{(d+e x)^{3/2} \left (\frac{5}{2} c (b d-2 a e)+\frac{5}{2} c (2 c d-b e) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{7 c}\\ &=\frac{2 (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{4 \int \frac{\sqrt{d+e x} \left (\frac{5}{4} c \left (b^2 d e-16 a c d e+3 b \left (c d^2+a e^2\right )\right )+\frac{5}{2} c \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{35 c^2}\\ &=\frac{4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{21 c^2}+\frac{2 (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{8 \int \frac{-\frac{5}{8} c \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )+\frac{5}{8} c (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3}\\ &=\frac{4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{21 c^2}+\frac{2 (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{\left ((2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{21 c^2 e}--\frac{\left (8 \left (-\frac{5}{8} c d (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac{5}{8} c e \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 e}\\ &=\frac{4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{21 c^2}+\frac{2 (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{21 c^3 e \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}--\frac{\left (16 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{5}{8} c d (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac{5}{8} c e \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{105 c^4 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{21 c^2}+\frac{2 (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{4}{7} (d+e x)^{5/2} \sqrt{a+b x+c x^2}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2-5 a c e^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 13.0243, size = 5339, normalized size = 8.9 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.053, size = 6517, normalized size = 10.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, c x + b\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (2 \, c e^{2} x^{3} + b d^{2} +{\left (4 \, c d e + b e^{2}\right )} x^{2} + 2 \,{\left (c d^{2} + b d e\right )} x\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((2*c*e^2*x^3 + b*d^2 + (4*c*d*e + b*e^2)*x^2 + 2*(c*d^2 + b*d*e)*x)*sqrt(e*x + d)/sqrt(c*x^2 + b*x +
a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**(5/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out