3.8.26 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [726]

Optimal. Leaf size=262 \[ -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {32 c^2 d^2 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-12/5*g*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^(5/2)/(e*x+d)^(1/2)-16/5*c*d*g*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)/(-a*e*g+c*d*f)^3/(g*x+f)^(3/2)/(e*x+d)^(1/2)-32/5*c^2*d^2*g*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(
-a*e*g+c*d*f)^4/(e*x+d)^(1/2)/(g*x+f)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {882, 886, 874} \begin {gather*} -\frac {32 c^2 d^2 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}-\frac {16 c d g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac {12 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (12*g*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(5/2)) - (16*c*d*g*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) - (32*c^2*d^2*g*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*Sqrt[f + g*x])

Rule 874

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))),
x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d
*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]

Rule 882

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x]
+ Dist[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*
x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[
c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]

Rule 886

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))),
x] - Dist[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))), Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c
*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*
d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && LtQ[n, -1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(6 g) \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d f-a e g}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {(24 c d g) \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 (c d f-a e g)^2}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {\left (16 c^2 d^2 g\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 (c d f-a e g)^3}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {32 c^2 d^2 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 139, normalized size = 0.53 \begin {gather*} -\frac {2 (a e+c d x)^4 (d+e x)^{3/2} \left (g^3-\frac {5 c d g^2 (f+g x)}{a e+c d x}+\frac {15 c^2 d^2 g (f+g x)^2}{(a e+c d x)^2}+\frac {5 c^3 d^3 (f+g x)^3}{(a e+c d x)^3}\right )}{5 (c d f-a e g)^4 ((a e+c d x) (d+e x))^{3/2} (f+g x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*(a*e + c*d*x)^4*(d + e*x)^(3/2)*(g^3 - (5*c*d*g^2*(f + g*x))/(a*e + c*d*x) + (15*c^2*d^2*g*(f + g*x)^2)/(a
*e + c*d*x)^2 + (5*c^3*d^3*(f + g*x)^3)/(a*e + c*d*x)^3))/(5*(c*d*f - a*e*g)^4*((a*e + c*d*x)*(d + e*x))^(3/2)
*(f + g*x)^(5/2))

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Maple [A]
time = 0.14, size = 192, normalized size = 0.73

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} c^{3} d^{3}\right )}{5 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (c d x +a e \right ) \left (a e g -c d f \right )^{4}}\)	\(192\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(259\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5/(e*x+d)^(1/2)/(g*x+f)^(5/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(16*c^3*d^3*g^3*x^3+8*a*c^2*d^2*e*g^3*x^2+40*c^3*
d^3*f*g^2*x^2-2*a^2*c*d*e^2*g^3*x+20*a*c^2*d^2*e*f*g^2*x+30*c^3*d^3*f^2*g*x+a^3*e^3*g^3-5*a^2*c*d*e^2*f*g^2+15
*a*c^2*d^2*e*f^2*g+5*c^3*d^3*f^3)/(c*d*x+a*e)/(a*e*g-c*d*f)^4

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1114 vs. \(2 (244) = 488\).
time = 3.76, size = 1114, normalized size = 4.25 \begin {gather*} -\frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} + 40 \, c^{3} d^{3} f g^{2} x^{2} + 30 \, c^{3} d^{3} f^{2} g x + 5 \, c^{3} d^{3} f^{3} + a^{3} g^{3} e^{3} - {\left (2 \, a^{2} c d g^{3} x + 5 \, a^{2} c d f g^{2}\right )} e^{2} + {\left (8 \, a c^{2} d^{2} g^{3} x^{2} + 20 \, a c^{2} d^{2} f g^{2} x + 15 \, a c^{2} d^{2} f^{2} g\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{5 \, {\left (c^{5} d^{6} f^{4} g^{3} x^{4} + 3 \, c^{5} d^{6} f^{5} g^{2} x^{3} + 3 \, c^{5} d^{6} f^{6} g x^{2} + c^{5} d^{6} f^{7} x + {\left (a^{5} g^{7} x^{4} + 3 \, a^{5} f g^{6} x^{3} + 3 \, a^{5} f^{2} g^{5} x^{2} + a^{5} f^{3} g^{4} x\right )} e^{6} + {\left (a^{4} c d g^{7} x^{5} - a^{4} c d f g^{6} x^{4} + a^{5} d f^{3} g^{4} - {\left (9 \, a^{4} c d f^{2} g^{5} - a^{5} d g^{7}\right )} x^{3} - {\left (11 \, a^{4} c d f^{3} g^{4} - 3 \, a^{5} d f g^{6}\right )} x^{2} - {\left (4 \, a^{4} c d f^{4} g^{3} - 3 \, a^{5} d f^{2} g^{5}\right )} x\right )} e^{5} - {\left (4 \, a^{3} c^{2} d^{2} f g^{6} x^{5} + 4 \, a^{4} c d^{2} f^{4} g^{3} + {\left (6 \, a^{3} c^{2} d^{2} f^{2} g^{5} - a^{4} c d^{2} g^{7}\right )} x^{4} - {\left (6 \, a^{3} c^{2} d^{2} f^{3} g^{4} - a^{4} c d^{2} f g^{6}\right )} x^{3} - {\left (14 \, a^{3} c^{2} d^{2} f^{4} g^{3} - 9 \, a^{4} c d^{2} f^{2} g^{5}\right )} x^{2} - {\left (6 \, a^{3} c^{2} d^{2} f^{5} g^{2} - 11 \, a^{4} c d^{2} f^{3} g^{4}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{3} d^{3} f^{2} g^{5} x^{5} + 3 \, a^{3} c^{2} d^{3} f^{5} g^{2} + {\left (7 \, a^{2} c^{3} d^{3} f^{3} g^{4} - 2 \, a^{3} c^{2} d^{3} f g^{6}\right )} x^{4} + 3 \, {\left (a^{2} c^{3} d^{3} f^{4} g^{3} - a^{3} c^{2} d^{3} f^{2} g^{5}\right )} x^{3} - 3 \, {\left (a^{2} c^{3} d^{3} f^{5} g^{2} - a^{3} c^{2} d^{3} f^{3} g^{4}\right )} x^{2} - {\left (2 \, a^{2} c^{3} d^{3} f^{6} g - 7 \, a^{3} c^{2} d^{3} f^{4} g^{3}\right )} x\right )} e^{3} - {\left (4 \, a c^{4} d^{4} f^{3} g^{4} x^{5} + 4 \, a^{2} c^{3} d^{4} f^{6} g + {\left (11 \, a c^{4} d^{4} f^{4} g^{3} - 6 \, a^{2} c^{3} d^{4} f^{2} g^{5}\right )} x^{4} + {\left (9 \, a c^{4} d^{4} f^{5} g^{2} - 14 \, a^{2} c^{3} d^{4} f^{3} g^{4}\right )} x^{3} + {\left (a c^{4} d^{4} f^{6} g - 6 \, a^{2} c^{3} d^{4} f^{4} g^{3}\right )} x^{2} - {\left (a c^{4} d^{4} f^{7} - 6 \, a^{2} c^{3} d^{4} f^{5} g^{2}\right )} x\right )} e^{2} + {\left (c^{5} d^{5} f^{4} g^{3} x^{5} - a c^{4} d^{5} f^{6} g x + a c^{4} d^{5} f^{7} + {\left (3 \, c^{5} d^{5} f^{5} g^{2} - 4 \, a c^{4} d^{5} f^{3} g^{4}\right )} x^{4} + {\left (3 \, c^{5} d^{5} f^{6} g - 11 \, a c^{4} d^{5} f^{4} g^{3}\right )} x^{3} + {\left (c^{5} d^{5} f^{7} - 9 \, a c^{4} d^{5} f^{5} g^{2}\right )} x^{2}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(16*c^3*d^3*g^3*x^3 + 40*c^3*d^3*f*g^2*x^2 + 30*c^3*d^3*f^2*g*x + 5*c^3*d^3*f^3 + a^3*g^3*e^3 - (2*a^2*c*
d*g^3*x + 5*a^2*c*d*f*g^2)*e^2 + (8*a*c^2*d^2*g^3*x^2 + 20*a*c^2*d^2*f*g^2*x + 15*a*c^2*d^2*f^2*g)*e)*sqrt(c*d
^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)*sqrt(x*e + d)/(c^5*d^6*f^4*g^3*x^4 + 3*c^5*d^6*f^5*g^2*x^3 +
3*c^5*d^6*f^6*g*x^2 + c^5*d^6*f^7*x + (a^5*g^7*x^4 + 3*a^5*f*g^6*x^3 + 3*a^5*f^2*g^5*x^2 + a^5*f^3*g^4*x)*e^6
+ (a^4*c*d*g^7*x^5 - a^4*c*d*f*g^6*x^4 + a^5*d*f^3*g^4 - (9*a^4*c*d*f^2*g^5 - a^5*d*g^7)*x^3 - (11*a^4*c*d*f^
3*g^4 - 3*a^5*d*f*g^6)*x^2 - (4*a^4*c*d*f^4*g^3 - 3*a^5*d*f^2*g^5)*x)*e^5 - (4*a^3*c^2*d^2*f*g^6*x^5 + 4*a^4*c
*d^2*f^4*g^3 + (6*a^3*c^2*d^2*f^2*g^5 - a^4*c*d^2*g^7)*x^4 - (6*a^3*c^2*d^2*f^3*g^4 - a^4*c*d^2*f*g^6)*x^3 - (
14*a^3*c^2*d^2*f^4*g^3 - 9*a^4*c*d^2*f^2*g^5)*x^2 - (6*a^3*c^2*d^2*f^5*g^2 - 11*a^4*c*d^2*f^3*g^4)*x)*e^4 + 2*
(3*a^2*c^3*d^3*f^2*g^5*x^5 + 3*a^3*c^2*d^3*f^5*g^2 + (7*a^2*c^3*d^3*f^3*g^4 - 2*a^3*c^2*d^3*f*g^6)*x^4 + 3*(a^
2*c^3*d^3*f^4*g^3 - a^3*c^2*d^3*f^2*g^5)*x^3 - 3*(a^2*c^3*d^3*f^5*g^2 - a^3*c^2*d^3*f^3*g^4)*x^2 - (2*a^2*c^3*
d^3*f^6*g - 7*a^3*c^2*d^3*f^4*g^3)*x)*e^3 - (4*a*c^4*d^4*f^3*g^4*x^5 + 4*a^2*c^3*d^4*f^6*g + (11*a*c^4*d^4*f^4
*g^3 - 6*a^2*c^3*d^4*f^2*g^5)*x^4 + (9*a*c^4*d^4*f^5*g^2 - 14*a^2*c^3*d^4*f^3*g^4)*x^3 + (a*c^4*d^4*f^6*g - 6*
a^2*c^3*d^4*f^4*g^3)*x^2 - (a*c^4*d^4*f^7 - 6*a^2*c^3*d^4*f^5*g^2)*x)*e^2 + (c^5*d^5*f^4*g^3*x^5 - a*c^4*d^5*f
^6*g*x + a*c^4*d^5*f^7 + (3*c^5*d^5*f^5*g^2 - 4*a*c^4*d^5*f^3*g^4)*x^4 + (3*c^5*d^5*f^6*g - 11*a*c^4*d^5*f^4*g
^3)*x^3 + (c^5*d^5*f^7 - 9*a*c^4*d^5*f^5*g^2)*x^2)*e)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [B]
time = 5.70, size = 414, normalized size = 1.58 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,x\,\sqrt {d+e\,x}\,\left (-a^2\,e^2\,g^2+10\,a\,c\,d\,e\,f\,g+15\,c^2\,d^2\,f^2\right )}{5\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {\sqrt {d+e\,x}\,\left (\frac {2\,a^3\,e^3\,g^3}{5}-2\,a^2\,c\,d\,e^2\,f\,g^2+6\,a\,c^2\,d^2\,e\,f^2\,g+2\,c^3\,d^3\,f^3\right )}{c\,d\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c^2\,d^2\,g\,x^3\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c\,d\,x^2\,\left (a\,e\,g+5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {a\,f^2\,\sqrt {f+g\,x}}{c\,g^2}+\frac {x^2\,\sqrt {f+g\,x}\,\left (2\,c\,d^2\,f\,g+c\,d\,e\,f^2+a\,d\,e\,g^2+2\,a\,e^2\,f\,g\right )}{c\,d\,e\,g^2}+\frac {x^3\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+2\,c\,f\,d\,e+a\,g\,e^2\right )}{c\,d\,e\,g}+\frac {f\,x\,\sqrt {f+g\,x}\,\left (c\,f\,d^2+2\,a\,g\,d\,e+a\,f\,e^2\right )}{c\,d\,e\,g^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*x*(d + e*x)^(1/2)*(15*c^2*d^2*f^2 - a^2*e^2*g^2 + 10*a*c*d
*e*f*g))/(5*e*g*(a*e*g - c*d*f)^4) + ((d + e*x)^(1/2)*((2*a^3*e^3*g^3)/5 + 2*c^3*d^3*f^3 + 6*a*c^2*d^2*e*f^2*g
- 2*a^2*c*d*e^2*f*g^2))/(c*d*e*g^2*(a*e*g - c*d*f)^4) + (32*c^2*d^2*g*x^3*(d + e*x)^(1/2))/(5*e*(a*e*g - c*d*
f)^4) + (16*c*d*x^2*(a*e*g + 5*c*d*f)*(d + e*x)^(1/2))/(5*e*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^(1/2) + (a*f^2
*(f + g*x)^(1/2))/(c*g^2) + (x^2*(f + g*x)^(1/2)*(a*d*e*g^2 + c*d*e*f^2 + 2*a*e^2*f*g + 2*c*d^2*f*g))/(c*d*e*g
^2) + (x^3*(f + g*x)^(1/2)*(a*e^2*g + c*d^2*g + 2*c*d*e*f))/(c*d*e*g) + (f*x*(f + g*x)^(1/2)*(a*e^2*f + c*d^2*
f + 2*a*d*e*g))/(c*d*e*g^2))

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