3.666 \(\int \frac{(d+e x)^{7/2}}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=413 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (71 c d^2-25 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (11 c d^2-13 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} \sqrt{d+e x} \left (71 c d^2-25 a e^2\right )}{105 c^2}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{5/2}}{7 c}+\frac{24 d e \sqrt{a+c x^2} (d+e x)^{3/2}}{35 c} \]

[Out]

(2*e*(71*c*d^2 - 25*a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(105*c^2) + (24*d*e*(d
+ e*x)^(3/2)*Sqrt[a + c*x^2])/(35*c) + (2*e*(d + e*x)^(5/2)*Sqrt[a + c*x^2])/(7
*c) - (32*Sqrt[-a]*d*(11*c*d^2 - 13*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ell
ipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c
]*d - a*e)])/(105*c^(3/2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqr
t[a + c*x^2]) + (2*Sqrt[-a]*(71*c*d^2 - 25*a*e^2)*(c*d^2 + a*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)])/(105*c^
(5/2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.24786, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (71 c d^2-25 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (11 c d^2-13 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} \sqrt{d+e x} \left (71 c d^2-25 a e^2\right )}{105 c^2}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{5/2}}{7 c}+\frac{24 d e \sqrt{a+c x^2} (d+e x)^{3/2}}{35 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/Sqrt[a + c*x^2],x]

[Out]

(2*e*(71*c*d^2 - 25*a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(105*c^2) + (24*d*e*(d
+ e*x)^(3/2)*Sqrt[a + c*x^2])/(35*c) + (2*e*(d + e*x)^(5/2)*Sqrt[a + c*x^2])/(7
*c) - (32*Sqrt[-a]*d*(11*c*d^2 - 13*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ell
ipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c
]*d - a*e)])/(105*c^(3/2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqr
t[a + c*x^2]) + (2*Sqrt[-a]*(71*c*d^2 - 25*a*e^2)*(c*d^2 + a*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)])/(105*c^
(5/2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(c*x**2+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 5.02807, size = 548, normalized size = 1.33 \[ \frac{2 \sqrt{d+e x} \left (\frac{16 d e \left (-13 a^2 e^2+a c \left (11 d^2-13 e^2 x^2\right )+11 c^2 d^2 x^2\right )}{d+e x}+\frac{\sqrt{d+e x} \left (208 a^{3/2} \sqrt{c} d e^3+25 i a^2 e^4-176 \sqrt{a} c^{3/2} d^3 e-254 i a c d^2 e^2+105 i c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{16 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (11 c d^2-13 a e^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e}+\left (a+c x^2\right ) \left (c e \left (122 d^2+66 d e x+15 e^2 x^2\right )-25 a e^3\right )\right )}{105 c^2 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/Sqrt[a + c*x^2],x]

[Out]

(2*Sqrt[d + e*x]*((16*d*e*(-13*a^2*e^2 + 11*c^2*d^2*x^2 + a*c*(11*d^2 - 13*e^2*x
^2)))/(d + e*x) + (a + c*x^2)*(-25*a*e^3 + c*e*(122*d^2 + 66*d*e*x + 15*e^2*x^2)
) + ((16*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(11*c*d^2 - 13*a*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))]*Sqrt[d + e*x]*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)])/e + (((105*I)*c^2
*d^4 - 176*Sqrt[a]*c^(3/2)*d^3*e - (254*I)*a*c*d^2*e^2 + 208*a^(3/2)*Sqrt[c]*d*e
^3 + (25*I)*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sq
rt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*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*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(105*c^2*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.052, size = 1534, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(c*x^2+a)^(1/2),x)

[Out]

-2/105*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(-15*x^5*c^3*e^5+25*(-a*c)^(1/2)*(-(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*e^5
-46*(-a*c)^(1/2)*(-(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)*El
lipticF((-(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*d^2*e^3-71*(-a*c)^(1/2)*(-(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))*c^2*d^4*e+183*a^2*c*(-(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))*d*e^4+
78*a*c^2*(-(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))*d^3*e^2-105*(-(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))*c^3*d^5-208*a^2*c*(-(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))*d*e^4-32*a*c^2*(-(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))*d^3*e^2+176*(-
(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^5-81*x^4*c^3*d*e^4+10*x^3*a*c^2*e^5-188*x^3*c^3*d^2*e^3-56*x^2*a*c^2*d*e^
4-122*x^2*c^3*d^3*e^2+25*x*a^2*c*e^5-188*x*a*c^2*d^2*e^3+25*a^2*c*d*e^4-122*a*c^
2*d^3*e^2)/e/(c*e*x^3+c*d*x^2+a*e*x+a*d)/c^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(c*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/sqrt(c*x^2 + a), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(c*x^2 + a),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(7/2)/(c*x**2+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(c*x^2 + a),x, algorithm="giac")

[Out]

Exception raised: RuntimeError