3.605 \(\int \frac{(d+e x)^{3/2} \sqrt{f+g x}}{a+c x^2} \, dx\)

Optimal. Leaf size=411 \[ \frac{\left (\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}-\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{a c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\left (\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )+\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{e \sqrt{d+e x} \sqrt{f+g x}}{c}+\frac{\sqrt{e} (3 d g+e f) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c \sqrt{g}} \]

[Out]

(e*Sqrt[d + e*x]*Sqrt[f + g*x])/c + (Sqrt[e]*(e*f + 3*d*g)*ArcTanh[(Sqrt[g]*Sqrt
[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(c*Sqrt[g]) + (((a*(a*e^2*g - c*d*(2*e*f +
d*g)))/Sqrt[c] - Sqrt[-a]*(c*d^2*f - a*e*(e*f + 2*d*g)))*ArcTanh[(Sqrt[Sqrt[c]*f
- Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*
c*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) + (((a*(a*e^2*g - c
*d*(2*e*f + d*g)))/Sqrt[c] + Sqrt[-a]*(c*d^2*f - a*e*(e*f + 2*d*g)))*ArcTanh[(Sq
rt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f +
g*x])])/(a*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

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Rubi [A]  time = 4.64825, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\left (\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}-\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{a c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\left (\sqrt{-a} \left (c d^2 f-a e (2 d g+e f)\right )+\frac{a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt{c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{e \sqrt{d+e x} \sqrt{f+g x}}{c}+\frac{\sqrt{e} (3 d g+e f) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c \sqrt{g}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(3/2)*Sqrt[f + g*x])/(a + c*x^2),x]

[Out]

(e*Sqrt[d + e*x]*Sqrt[f + g*x])/c + (Sqrt[e]*(e*f + 3*d*g)*ArcTanh[(Sqrt[g]*Sqrt
[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(c*Sqrt[g]) + (((a*(a*e^2*g - c*d*(2*e*f +
d*g)))/Sqrt[c] - Sqrt[-a]*(c*d^2*f - a*e*(e*f + 2*d*g)))*ArcTanh[(Sqrt[Sqrt[c]*f
- Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*
c*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) + (((a*(a*e^2*g - c
*d*(2*e*f + d*g)))/Sqrt[c] + Sqrt[-a]*(c*d^2*f - a*e*(e*f + 2*d*g)))*ArcTanh[(Sq
rt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f +
g*x])])/(a*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

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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)**(3/2)*(g*x+f)**(1/2)/(c*x**2+a),x)

[Out]

Timed out

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Mathematica [C]  time = 8.30699, size = 1076, normalized size = 2.62 \[ \frac{\sqrt{d+e x} \sqrt{f+g x} e}{c}+\frac{(e f+3 d g) \log \left (e f+d g+2 e g x+2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x}\right ) \sqrt{e}}{2 c \sqrt{g}}+\frac{\left (-i c^2 f d^2-\sqrt{a} c^{3/2} g d^2-2 \sqrt{a} c^{3/2} e f d+2 i a c e g d+i a c e^2 f+a^{3/2} \sqrt{c} e^2 g\right ) \log \left (\frac{2 i \sqrt{a} \sqrt{d+e x} \sqrt{f+g x} c^{5/2}}{\left (-i c^2 f d^2-\sqrt{a} c^{3/2} g d^2-2 \sqrt{a} c^{3/2} e f d+2 i a c e g d+i a c e^2 f+a^{3/2} \sqrt{c} e^2 g\right ) \left (\sqrt{a} \sqrt{c}-i c x\right )}+\frac{i \left (2 \sqrt{a} d f c^{5/2}+\sqrt{a} e f x c^{5/2}+\sqrt{a} d g x c^{5/2}-i a e f c^2-i a d g c^2-2 i a e g x c^2\right ) \sqrt{c}}{\sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g} \left (-i c^2 f d^2-\sqrt{a} c^{3/2} g d^2-2 \sqrt{a} c^{3/2} e f d+2 i a c e g d+i a c e^2 f+a^{3/2} \sqrt{c} e^2 g\right ) \left (\sqrt{a} \sqrt{c}-i c x\right )}\right )}{2 \sqrt{a} c^2 \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}}+\frac{\left (i c^2 f d^2-\sqrt{a} c^{3/2} g d^2-2 \sqrt{a} c^{3/2} e f d-2 i a c e g d-i a c e^2 f+a^{3/2} \sqrt{c} e^2 g\right ) \log \left (-\frac{2 i \sqrt{a} \sqrt{d+e x} \sqrt{f+g x} c^{5/2}}{\left (i c^2 f d^2-\sqrt{a} c^{3/2} g d^2-2 \sqrt{a} c^{3/2} e f d-2 i a c e g d-i a c e^2 f+a^{3/2} \sqrt{c} e^2 g\right ) \left (i c x+\sqrt{a} \sqrt{c}\right )}-\frac{i \left (2 \sqrt{a} d f c^{5/2}+\sqrt{a} e f x c^{5/2}+\sqrt{a} d g x c^{5/2}+i a e f c^2+i a d g c^2+2 i a e g x c^2\right ) \sqrt{c}}{\sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g} \left (i c^2 f d^2-\sqrt{a} c^{3/2} g d^2-2 \sqrt{a} c^{3/2} e f d-2 i a c e g d-i a c e^2 f+a^{3/2} \sqrt{c} e^2 g\right ) \left (i c x+\sqrt{a} \sqrt{c}\right )}\right )}{2 \sqrt{a} c^2 \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(3/2)*Sqrt[f + g*x])/(a + c*x^2),x]

[Out]

(e*Sqrt[d + e*x]*Sqrt[f + g*x])/c + (Sqrt[e]*(e*f + 3*d*g)*Log[e*f + d*g + 2*e*g
*x + 2*Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]*Sqrt[f + g*x]])/(2*c*Sqrt[g]) + (((-I)*c^2*
d^2*f - 2*Sqrt[a]*c^(3/2)*d*e*f + I*a*c*e^2*f - Sqrt[a]*c^(3/2)*d^2*g + (2*I)*a*
c*d*e*g + a^(3/2)*Sqrt[c]*e^2*g)*Log[((2*I)*Sqrt[a]*c^(5/2)*Sqrt[d + e*x]*Sqrt[f
+ g*x])/(((-I)*c^2*d^2*f - 2*Sqrt[a]*c^(3/2)*d*e*f + I*a*c*e^2*f - Sqrt[a]*c^(3
/2)*d^2*g + (2*I)*a*c*d*e*g + a^(3/2)*Sqrt[c]*e^2*g)*(Sqrt[a]*Sqrt[c] - I*c*x))
+ (I*Sqrt[c]*(2*Sqrt[a]*c^(5/2)*d*f - I*a*c^2*e*f - I*a*c^2*d*g + Sqrt[a]*c^(5/2
)*e*f*x + Sqrt[a]*c^(5/2)*d*g*x - (2*I)*a*c^2*e*g*x))/(Sqrt[Sqrt[c]*d - I*Sqrt[a
]*e]*Sqrt[Sqrt[c]*f - I*Sqrt[a]*g]*((-I)*c^2*d^2*f - 2*Sqrt[a]*c^(3/2)*d*e*f + I
*a*c*e^2*f - Sqrt[a]*c^(3/2)*d^2*g + (2*I)*a*c*d*e*g + a^(3/2)*Sqrt[c]*e^2*g)*(S
qrt[a]*Sqrt[c] - I*c*x))])/(2*Sqrt[a]*c^2*Sqrt[Sqrt[c]*d - I*Sqrt[a]*e]*Sqrt[Sqr
t[c]*f - I*Sqrt[a]*g]) + ((I*c^2*d^2*f - 2*Sqrt[a]*c^(3/2)*d*e*f - I*a*c*e^2*f -
Sqrt[a]*c^(3/2)*d^2*g - (2*I)*a*c*d*e*g + a^(3/2)*Sqrt[c]*e^2*g)*Log[((-2*I)*Sq
rt[a]*c^(5/2)*Sqrt[d + e*x]*Sqrt[f + g*x])/((I*c^2*d^2*f - 2*Sqrt[a]*c^(3/2)*d*e
*f - I*a*c*e^2*f - Sqrt[a]*c^(3/2)*d^2*g - (2*I)*a*c*d*e*g + a^(3/2)*Sqrt[c]*e^2
*g)*(Sqrt[a]*Sqrt[c] + I*c*x)) - (I*Sqrt[c]*(2*Sqrt[a]*c^(5/2)*d*f + I*a*c^2*e*f
+ I*a*c^2*d*g + Sqrt[a]*c^(5/2)*e*f*x + Sqrt[a]*c^(5/2)*d*g*x + (2*I)*a*c^2*e*g
*x))/(Sqrt[Sqrt[c]*d + I*Sqrt[a]*e]*Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]*(I*c^2*d^2*f -
2*Sqrt[a]*c^(3/2)*d*e*f - I*a*c*e^2*f - Sqrt[a]*c^(3/2)*d^2*g - (2*I)*a*c*d*e*g
+ a^(3/2)*Sqrt[c]*e^2*g)*(Sqrt[a]*Sqrt[c] + I*c*x))])/(2*Sqrt[a]*c^2*Sqrt[Sqrt[
c]*d + I*Sqrt[a]*e]*Sqrt[Sqrt[c]*f + I*Sqrt[a]*g])

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Maple [B]  time = 0.153, size = 2497, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(g*x+f)^(1/2)/(c*x^2+a),x)

[Out]

1/2*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(ln((2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+(-a*c)
^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(((-a*c)^(1/2)*d*g
+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+2*c*d*f)/(c*x-(-a*c)^(1/2)))*a*e^2*g*(
e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(-a*c)^(1/
2)-ln((2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*
(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)
/c)^(1/2)*c+2*c*d*f)/(c*x-(-a*c)^(1/2)))*d^2*g*c*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g
+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(-a*c)^(1/2)-2*ln((2*(-a*c)^(1/2)*x*e*g+
x*c*d*g+x*c*e*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1
/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+2*c*d*f)/(c*x-(-
a*c)^(1/2)))*e*f*d*c*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*
f)/c)^(1/2)*(-a*c)^(1/2)+2*a*c*ln((2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+(-a*c)^(
1/2)*d*g+(-a*c)^(1/2)*e*f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(((-a*c)^(1/2)*d*g+(
-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+2*c*d*f)/(c*x-(-a*c)^(1/2)))*d*g*e*(e*g)
^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)+a*c*ln((2*(-a*
c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*(e*g*x^2+d*g*
x+e*f*x+d*f)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+2
*c*d*f)/(c*x-(-a*c)^(1/2)))*e^2*f*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e
*f+a*e*g-c*d*f)/c)^(1/2)-ln((2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+(-a*c)^(1/2)*d
*g+(-a*c)^(1/2)*e*f+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^
(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c+2*c*d*f)/(c*x-(-a*c)^(1/2)))*d^2*f*c^2*(e*g)^(
1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)+ln((-2*(-a*c)^(1
/2)*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)
^(1/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d
*f)/(c*x+(-a*c)^(1/2)))*a*e^2*g*(e*g)^(1/2)*(-a*c)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*
c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)-ln((-2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(
-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(e*g*x^2+d*g*x+e*f*x+d
*f)^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*d^2*g
*c*(e*g)^(1/2)*(-a*c)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^
(1/2)-2*ln((-2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(
1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/2)*d*
g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*e*f*d*c*(e*g)^(1/2)*(-a*c)^(1/2)
*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)-2*a*c*ln((-2*(-a*c)^(
1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c
)^(1/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*
d*f)/(c*x+(-a*c)^(1/2)))*d*g*e*(e*g)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a
*e*g+c*d*f)/c)^(1/2)-a*c*ln((-2*(-a*c)^(1/2)*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a*c)^(
1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*
c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*e^2*f*(e*g)^(1/
2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)+ln((-2*(-a*c)^(1/2)
*x*e*g+x*c*d*g+x*c*e*f+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1
/2)*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)
/(c*x+(-a*c)^(1/2)))*d^2*f*c^2*(e*g)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a
*e*g+c*d*f)/c)^(1/2)+3*ln(1/2*(2*e*g*x+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(e*g)^(
1/2)+d*g+e*f)/(e*g)^(1/2))*e*g*d*c*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*
d*f)/c)^(1/2)*(-a*c)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(
1/2)+ln(1/2*(2*e*g*x+2*(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g
)^(1/2))*e^2*f*c*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*(-a*
c)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)+2*(e*g*x^2+d*
g*x+e*f*x+d*f)^(1/2)*e*c*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-
c*d*f)/c)^(1/2)*(-a*c)^(1/2)*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)
^(1/2))/(e*g*x^2+d*g*x+e*f*x+d*f)^(1/2)/c^2/(e*g)^(1/2)/(-((-a*c)^(1/2)*d*g+(-a*
c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)/(-a*c)^(1/2)/(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*
e*f-a*e*g+c*d*f)/c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*sqrt(g*x + f)/(c*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)*sqrt(g*x + f)/(c*x^2 + a), x)

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Fricas [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)^(3/2)*sqrt(g*x + f)/(c*x^2 + a),x, algorithm="fricas")

[Out]

Timed out

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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)**(3/2)*(g*x+f)**(1/2)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 3.55089, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*sqrt(g*x + f)/(c*x^2 + a),x, algorithm="giac")

[Out]

sage0*x