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

Optimal. Leaf size=214 \[ -\frac{\left (15 a e^2 g^2+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}}-\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{2 e (d+e x)^2 (e f-d g)^2}+\frac{\sqrt{f+g x} \left (7 a e^2 g+c d (8 e f-d g)\right )}{4 e (d+e x) (e f-d g)^3}+\frac{2 \left (a g^2+c f^2\right )}{\sqrt{f+g x} (e f-d g)^3} \]

[Out]

(2*(c*f^2 + a*g^2))/((e*f - d*g)^3*Sqrt[f + g*x]) - ((c*d^2 + a*e^2)*Sqrt[f + g*x])/(2*e*(e*f - d*g)^2*(d + e*
x)^2) + ((7*a*e^2*g + c*d*(8*e*f - d*g))*Sqrt[f + g*x])/(4*e*(e*f - d*g)^3*(d + e*x)) - ((15*a*e^2*g^2 + c*(8*
e^2*f^2 + 8*d*e*f*g - d^2*g^2))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(4*e^(3/2)*(e*f - d*g)^(7/2)
)

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Rubi [A]  time = 0.504543, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {898, 1259, 456, 453, 208} \[ -\frac{\left (15 a e^2 g^2+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}}-\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{2 e (d+e x)^2 (e f-d g)^2}+\frac{\sqrt{f+g x} \left (7 a e^2 g+c d (8 e f-d g)\right )}{4 e (d+e x) (e f-d g)^3}+\frac{2 \left (a g^2+c f^2\right )}{\sqrt{f+g x} (e f-d g)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*f^2 + a*g^2))/((e*f - d*g)^3*Sqrt[f + g*x]) - ((c*d^2 + a*e^2)*Sqrt[f + g*x])/(2*e*(e*f - d*g)^2*(d + e*
x)^2) + ((7*a*e^2*g + c*d*(8*e*f - d*g))*Sqrt[f + g*x])/(4*e*(e*f - d*g)^3*(d + e*x)) - ((15*a*e^2*g^2 + c*(8*
e^2*f^2 + 8*d*e*f*g - d^2*g^2))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(4*e^(3/2)*(e*f - d*g)^(7/2)
)

Rule 898

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c
*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{a+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\frac{c f^2+a g^2}{g^2}-\frac{2 c f x^2}{g^2}+\frac{c x^4}{g^2}}{x^2 \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^3} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=-\frac{\left (c d^2+a e^2\right ) \sqrt{f+g x}}{2 e (e f-d g)^2 (d+e x)^2}-\frac{g^3 \operatorname{Subst}\left (\int \frac{\frac{4 e^2 (e f-d g) \left (c f^2+a g^2\right )}{g^5}+\frac{e \left (3 a e^2 g^2-c \left (4 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) x^2}{g^5}}{x^2 \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^2} \, dx,x,\sqrt{f+g x}\right )}{2 e^2 (e f-d g)^2}\\ &=-\frac{\left (c d^2+a e^2\right ) \sqrt{f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac{\left (7 a e^2 g+c d (8 e f-d g)\right ) \sqrt{f+g x}}{4 e (e f-d g)^3 (d+e x)}+\frac{g^3 \operatorname{Subst}\left (\int \frac{\frac{8 e^2 \left (c f^2+a g^2\right )}{g^4}+\frac{e \left (7 a e^2 g+c d (8 e f-d g)\right ) x^2}{g^3 (e f-d g)}}{x^2 \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )} \, dx,x,\sqrt{f+g x}\right )}{4 e^2 (e f-d g)^2}\\ &=\frac{2 \left (c f^2+a g^2\right )}{(e f-d g)^3 \sqrt{f+g x}}-\frac{\left (c d^2+a e^2\right ) \sqrt{f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac{\left (7 a e^2 g+c d (8 e f-d g)\right ) \sqrt{f+g x}}{4 e (e f-d g)^3 (d+e x)}+\frac{\left (15 a e^2 g^2+c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{-e f+d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{4 e g (e f-d g)^3}\\ &=\frac{2 \left (c f^2+a g^2\right )}{(e f-d g)^3 \sqrt{f+g x}}-\frac{\left (c d^2+a e^2\right ) \sqrt{f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac{\left (7 a e^2 g+c d (8 e f-d g)\right ) \sqrt{f+g x}}{4 e (e f-d g)^3 (d+e x)}-\frac{\left (15 a e^2 g^2+c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.107233, size = 140, normalized size = 0.65 \[ \frac{2 \left (g \left (g \left (a e^2+c d^2\right ) \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )+2 c d (e f-d g) \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )\right )+c (e f-d g)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )\right )}{e^2 \sqrt{f+g x} (e f-d g)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*(e*f - d*g)^2*Hypergeometric2F1[-1/2, 1, 1/2, (e*(f + g*x))/(e*f - d*g)] + g*(2*c*d*(e*f - d*g)*Hypergeo
metric2F1[-1/2, 2, 1/2, (e*(f + g*x))/(e*f - d*g)] + (c*d^2 + a*e^2)*g*Hypergeometric2F1[-1/2, 3, 1/2, (e*(f +
g*x))/(e*f - d*g)])))/(e^2*(e*f - d*g)^3*Sqrt[f + g*x])

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Maple [B]  time = 0.224, size = 546, normalized size = 2.6 \begin{align*} -2\,{\frac{a{g}^{2}}{ \left ( dg-ef \right ) ^{3}\sqrt{gx+f}}}-2\,{\frac{c{f}^{2}}{ \left ( dg-ef \right ) ^{3}\sqrt{gx+f}}}-{\frac{7\,a{e}^{2}{g}^{2}}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}+{\frac{c{d}^{2}{g}^{2}}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}-2\,{\frac{ \left ( gx+f \right ) ^{3/2}cdefg}{ \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}}-{\frac{9\,{g}^{3}ead}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}\sqrt{gx+f}}+{\frac{9\,a{e}^{2}{g}^{2}f}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}\sqrt{gx+f}}-{\frac{c{d}^{3}{g}^{3}}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}e}\sqrt{gx+f}}-{\frac{7\,c{d}^{2}{g}^{2}f}{4\, \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}\sqrt{gx+f}}+2\,{\frac{eg\sqrt{gx+f}cd{f}^{2}}{ \left ( dg-ef \right ) ^{3} \left ( egx+dg \right ) ^{2}}}-{\frac{15\,ae{g}^{2}}{4\, \left ( dg-ef \right ) ^{3}}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}+{\frac{c{d}^{2}{g}^{2}}{4\, \left ( dg-ef \right ) ^{3}e}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}-2\,{\frac{cdfg}{ \left ( dg-ef \right ) ^{3}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-2\,{\frac{ce{f}^{2}}{ \left ( dg-ef \right ) ^{3}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(d*g-e*f)^3/(g*x+f)^(1/2)*a*g^2-2/(d*g-e*f)^3/(g*x+f)^(1/2)*c*f^2-7/4/(d*g-e*f)^3/(e*g*x+d*g)^2*(g*x+f)^(3/
2)*a*e^2*g^2+1/4/(d*g-e*f)^3/(e*g*x+d*g)^2*(g*x+f)^(3/2)*c*d^2*g^2-2/(d*g-e*f)^3/(e*g*x+d*g)^2*(g*x+f)^(3/2)*c
*d*e*f*g-9/4/(d*g-e*f)^3/(e*g*x+d*g)^2*g^3*e*(g*x+f)^(1/2)*a*d+9/4/(d*g-e*f)^3/(e*g*x+d*g)^2*g^2*e^2*(g*x+f)^(
1/2)*a*f-1/4/(d*g-e*f)^3/(e*g*x+d*g)^2*g^3/e*(g*x+f)^(1/2)*c*d^3-7/4/(d*g-e*f)^3/(e*g*x+d*g)^2*g^2*(g*x+f)^(1/
2)*c*d^2*f+2/(d*g-e*f)^3/(e*g*x+d*g)^2*g*e*(g*x+f)^(1/2)*c*d*f^2-15/4/(d*g-e*f)^3*e/((d*g-e*f)*e)^(1/2)*arctan
(e*(g*x+f)^(1/2)/((d*g-e*f)*e)^(1/2))*a*g^2+1/4/(d*g-e*f)^3/e/((d*g-e*f)*e)^(1/2)*arctan(e*(g*x+f)^(1/2)/((d*g
-e*f)*e)^(1/2))*c*d^2*g^2-2/(d*g-e*f)^3/((d*g-e*f)*e)^(1/2)*arctan(e*(g*x+f)^(1/2)/((d*g-e*f)*e)^(1/2))*c*d*f*
g-2/(d*g-e*f)^3*e/((d*g-e*f)*e)^(1/2)*arctan(e*(g*x+f)^(1/2)/((d*g-e*f)*e)^(1/2))*c*f^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.14663, size = 3135, normalized size = 14.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*((8*c*d^2*e^2*f^3 + 8*c*d^3*e*f^2*g - (c*d^4 - 15*a*d^2*e^2)*f*g^2 + (8*c*e^4*f^2*g + 8*c*d*e^3*f*g^2 -
(c*d^2*e^2 - 15*a*e^4)*g^3)*x^3 + (8*c*e^4*f^3 + 24*c*d*e^3*f^2*g + 15*(c*d^2*e^2 + a*e^4)*f*g^2 - 2*(c*d^3*e
- 15*a*d*e^3)*g^3)*x^2 + (16*c*d*e^3*f^3 + 24*c*d^2*e^2*f^2*g + 6*(c*d^3*e + 5*a*d*e^3)*f*g^2 - (c*d^4 - 15*a*
d^2*e^2)*g^3)*x)*sqrt(e^2*f - d*e*g)*log((e*g*x + 2*e*f - d*g + 2*sqrt(e^2*f - d*e*g)*sqrt(g*x + f))/(e*x + d)
) + 2*(8*a*d^3*e^2*g^3 - 2*(7*c*d^2*e^3 - a*e^5)*f^3 + (13*c*d^3*e^2 - 11*a*d*e^4)*f^2*g + (c*d^4*e + a*d^2*e^
3)*f*g^2 - (8*c*e^5*f^3 - 3*(3*c*d^2*e^3 - 5*a*e^5)*f*g^2 + (c*d^3*e^2 - 15*a*d*e^4)*g^3)*x^2 - (24*c*d*e^4*f^
3 - (19*c*d^2*e^3 - 5*a*e^5)*f^2*g - 4*(c*d^3*e^2 - 5*a*d*e^4)*f*g^2 - (c*d^4*e + 25*a*d^2*e^3)*g^3)*x)*sqrt(g
*x + f))/(d^2*e^6*f^5 - 4*d^3*e^5*f^4*g + 6*d^4*e^4*f^3*g^2 - 4*d^5*e^3*f^2*g^3 + d^6*e^2*f*g^4 + (e^8*f^4*g -
4*d*e^7*f^3*g^2 + 6*d^2*e^6*f^2*g^3 - 4*d^3*e^5*f*g^4 + d^4*e^4*g^5)*x^3 + (e^8*f^5 - 2*d*e^7*f^4*g - 2*d^2*e
^6*f^3*g^2 + 8*d^3*e^5*f^2*g^3 - 7*d^4*e^4*f*g^4 + 2*d^5*e^3*g^5)*x^2 + (2*d*e^7*f^5 - 7*d^2*e^6*f^4*g + 8*d^3
*e^5*f^3*g^2 - 2*d^4*e^4*f^2*g^3 - 2*d^5*e^3*f*g^4 + d^6*e^2*g^5)*x), 1/4*((8*c*d^2*e^2*f^3 + 8*c*d^3*e*f^2*g
- (c*d^4 - 15*a*d^2*e^2)*f*g^2 + (8*c*e^4*f^2*g + 8*c*d*e^3*f*g^2 - (c*d^2*e^2 - 15*a*e^4)*g^3)*x^3 + (8*c*e^4
*f^3 + 24*c*d*e^3*f^2*g + 15*(c*d^2*e^2 + a*e^4)*f*g^2 - 2*(c*d^3*e - 15*a*d*e^3)*g^3)*x^2 + (16*c*d*e^3*f^3 +
24*c*d^2*e^2*f^2*g + 6*(c*d^3*e + 5*a*d*e^3)*f*g^2 - (c*d^4 - 15*a*d^2*e^2)*g^3)*x)*sqrt(-e^2*f + d*e*g)*arct
an(sqrt(-e^2*f + d*e*g)*sqrt(g*x + f)/(e*g*x + e*f)) - (8*a*d^3*e^2*g^3 - 2*(7*c*d^2*e^3 - a*e^5)*f^3 + (13*c*
d^3*e^2 - 11*a*d*e^4)*f^2*g + (c*d^4*e + a*d^2*e^3)*f*g^2 - (8*c*e^5*f^3 - 3*(3*c*d^2*e^3 - 5*a*e^5)*f*g^2 + (
c*d^3*e^2 - 15*a*d*e^4)*g^3)*x^2 - (24*c*d*e^4*f^3 - (19*c*d^2*e^3 - 5*a*e^5)*f^2*g - 4*(c*d^3*e^2 - 5*a*d*e^4
)*f*g^2 - (c*d^4*e + 25*a*d^2*e^3)*g^3)*x)*sqrt(g*x + f))/(d^2*e^6*f^5 - 4*d^3*e^5*f^4*g + 6*d^4*e^4*f^3*g^2 -
4*d^5*e^3*f^2*g^3 + d^6*e^2*f*g^4 + (e^8*f^4*g - 4*d*e^7*f^3*g^2 + 6*d^2*e^6*f^2*g^3 - 4*d^3*e^5*f*g^4 + d^4*
e^4*g^5)*x^3 + (e^8*f^5 - 2*d*e^7*f^4*g - 2*d^2*e^6*f^3*g^2 + 8*d^3*e^5*f^2*g^3 - 7*d^4*e^4*f*g^4 + 2*d^5*e^3*
g^5)*x^2 + (2*d*e^7*f^5 - 7*d^2*e^6*f^4*g + 8*d^3*e^5*f^3*g^2 - 2*d^4*e^4*f^2*g^3 - 2*d^5*e^3*f*g^4 + d^6*e^2*
g^5)*x)]

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Sympy [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((c*x**2+a)/(e*x+d)**3/(g*x+f)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18016, size = 487, normalized size = 2.28 \begin{align*} \frac{{\left (c d^{2} g^{2} - 8 \, c d f g e - 8 \, c f^{2} e^{2} - 15 \, a g^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{4 \,{\left (d^{3} g^{3} e - 3 \, d^{2} f g^{2} e^{2} + 3 \, d f^{2} g e^{3} - f^{3} e^{4}\right )} \sqrt{d g e - f e^{2}}} - \frac{2 \,{\left (c f^{2} + a g^{2}\right )}}{{\left (d^{3} g^{3} - 3 \, d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} - f^{3} e^{3}\right )} \sqrt{g x + f}} - \frac{\sqrt{g x + f} c d^{3} g^{3} -{\left (g x + f\right )}^{\frac{3}{2}} c d^{2} g^{2} e + 7 \, \sqrt{g x + f} c d^{2} f g^{2} e + 8 \,{\left (g x + f\right )}^{\frac{3}{2}} c d f g e^{2} - 8 \, \sqrt{g x + f} c d f^{2} g e^{2} + 9 \, \sqrt{g x + f} a d g^{3} e^{2} + 7 \,{\left (g x + f\right )}^{\frac{3}{2}} a g^{2} e^{3} - 9 \, \sqrt{g x + f} a f g^{2} e^{3}}{4 \,{\left (d^{3} g^{3} e - 3 \, d^{2} f g^{2} e^{2} + 3 \, d f^{2} g e^{3} - f^{3} e^{4}\right )}{\left (d g +{\left (g x + f\right )} e - f e\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*d^2*g^2 - 8*c*d*f*g*e - 8*c*f^2*e^2 - 15*a*g^2*e^2)*arctan(sqrt(g*x + f)*e/sqrt(d*g*e - f*e^2))/((d^3*g
^3*e - 3*d^2*f*g^2*e^2 + 3*d*f^2*g*e^3 - f^3*e^4)*sqrt(d*g*e - f*e^2)) - 2*(c*f^2 + a*g^2)/((d^3*g^3 - 3*d^2*f
*g^2*e + 3*d*f^2*g*e^2 - f^3*e^3)*sqrt(g*x + f)) - 1/4*(sqrt(g*x + f)*c*d^3*g^3 - (g*x + f)^(3/2)*c*d^2*g^2*e
+ 7*sqrt(g*x + f)*c*d^2*f*g^2*e + 8*(g*x + f)^(3/2)*c*d*f*g*e^2 - 8*sqrt(g*x + f)*c*d*f^2*g*e^2 + 9*sqrt(g*x +
f)*a*d*g^3*e^2 + 7*(g*x + f)^(3/2)*a*g^2*e^3 - 9*sqrt(g*x + f)*a*f*g^2*e^3)/((d^3*g^3*e - 3*d^2*f*g^2*e^2 + 3
*d*f^2*g*e^3 - f^3*e^4)*(d*g + (g*x + f)*e - f*e)^2)