3.29.57 \(\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [2857]

Optimal. Leaf size=204 \[ -\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {f} \sqrt {-d e+c f} \sqrt {a+b x} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{(b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}} \sqrt {e+f x}} \]

[Out]

-2*b*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(1/2)+2*EllipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*
e)^(1/2),(-b*(-c*f+d*e)/(-a*d+b*c)/f)^(1/2))*f^(1/2)*(c*f-d*e)^(1/2)*(b*x+a)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2
)/(-a*d+b*c)/(-a*f+b*e)/(-d*(b*x+a)/(-a*d+b*c))^(1/2)/(f*x+e)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {106, 21, 115, 114} \begin {gather*} \frac {2 \sqrt {f} \sqrt {a+b x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\text {ArcSin}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}}}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d) (b e-a f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqr
t[a + b*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], -((b*
(d*e - c*f))/((b*c - a*d)*f))])/((b*c - a*d)*(b*e - a*f)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Sqrt[e + f*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 106

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 114

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

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \int \frac {-\frac {1}{2} a d f-\frac {1}{2} b d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {(d f) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {\left (d f \sqrt {a+b x} \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {\sqrt {\frac {a d}{-b c+a d}+\frac {b d x}{-b c+a d}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{(b c-a d) (b e-a f) \sqrt {\frac {d (a+b x)}{-b c+a d}} \sqrt {e+f x}}\\ &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {f} \sqrt {-d e+c f} \sqrt {a+b x} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{(b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}} \sqrt {e+f x}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 15.37, size = 201, normalized size = 0.99 \begin {gather*} \frac {2 b \sqrt {c+d x} \sqrt {e+f x} \left (-1-\frac {i \sqrt {\frac {d (a+b x)}{b (c+d x)}} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {d (a+b x)}{b c-a d}}\right )|\frac {b c f-a d f}{b d e-a d f}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {d (a+b x)}{b c-a d}}\right )|\frac {b c f-a d f}{b d e-a d f}\right )\right )}{\sqrt {\frac {b (e+f x)}{b e-a f}}}\right )}{(b c-a d) (b e-a f) \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*(-1 - (I*Sqrt[(d*(a + b*x))/(b*(c + d*x))]*(EllipticE[I*ArcSinh[Sqrt[(d*(a +
b*x))/(b*c - a*d)]], (b*c*f - a*d*f)/(b*d*e - a*d*f)] - EllipticF[I*ArcSinh[Sqrt[(d*(a + b*x))/(b*c - a*d)]],
(b*c*f - a*d*f)/(b*d*e - a*d*f)]))/Sqrt[(b*(e + f*x))/(b*e - a*f)]))/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(181)=362\).
time = 0.10, size = 565, normalized size = 2.77

method	result	size
default	\(-\frac {2 \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a c \,f^{2}-\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a d e f -\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b c e f +\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \EllipticE \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b d \,e^{2}+b d \,f^{2} x^{2}+b c \,f^{2} x +b d e f x +b c e f \right ) \sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {b x +a}}{f \left (a f -b e \right ) \left (a d -b c \right ) \left (b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e \right )}\)	\(565\)
elliptic	\(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right )}\, \left (-\frac {2 \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}{\left (a^{2} d f -a b c f -a b d e +b^{2} c e \right ) \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 \left (\frac {a d f -b c f -b d e}{a^{2} d f -a b c f -a b d e +b^{2} c e}+\frac {b c f +b d e}{a^{2} d f -a b c f -a b d e +b^{2} c e}\right ) \left (-\frac {c}{d}+\frac {e}{f}\right ) \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {2 b d f \left (-\frac {c}{d}+\frac {e}{f}\right ) \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a \EllipticF \left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{\left (a^{2} d f -a b c f -a b d e +b^{2} c e \right ) \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}\)	\(690\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*((-(f*x+e)*d/(c*f-d*e))^(1/2)*((b*x+a)*f/(a*f-b*e))^(1/2)*((d*x+c)*f/(c*f-d*e))^(1/2)*EllipticE((-(f*x+e)*d
/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*a*c*f^2-(-(f*x+e)*d/(c*f-d*e))^(1/2)*((b*x+a)*f/(a*f-b*e))^
(1/2)*((d*x+c)*f/(c*f-d*e))^(1/2)*EllipticE((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*a*d*
e*f-(-(f*x+e)*d/(c*f-d*e))^(1/2)*((b*x+a)*f/(a*f-b*e))^(1/2)*((d*x+c)*f/(c*f-d*e))^(1/2)*EllipticE((-(f*x+e)*d
/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*b*c*e*f+(-(f*x+e)*d/(c*f-d*e))^(1/2)*((b*x+a)*f/(a*f-b*e))^
(1/2)*((d*x+c)*f/(c*f-d*e))^(1/2)*EllipticE((-(f*x+e)*d/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*b*d*
e^2+b*d*f^2*x^2+b*c*f^2*x+b*d*e*f*x+b*c*e*f)*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(b*x+a)^(1/2)/f/(a*f-b*e)/(a*d-b*c)/(
b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)

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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(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.44, size = 813, normalized size = 3.99 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} b^{2} d f + \sqrt {b d f} {\left ({\left (b^{2} c - 2 \, a b d\right )} f x + {\left (a b c - 2 \, a^{2} d\right )} f + {\left (b^{2} d x + a b d\right )} e\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right ) + 3 \, {\left (b^{2} d f x + a b d f\right )} \sqrt {b d f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )\right )\right )}}{3 \, {\left ({\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} f^{2} x + {\left (a^{2} b^{2} c d - a^{3} b d^{2}\right )} f^{2} - {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} f x + {\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} f\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

2/3*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*b^2*d*f + sqrt(b*d*f)*((b^2*c - 2*a*b*d)*f*x + (a*b*c - 2*a^2
*d)*f + (b^2*d*x + a*b*d)*e)*weierstrassPInverse(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2 - (b^2*c
*d + a*b*d^2)*f*e)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^
3)*f^3 - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)/(b^3*d^3*f^3), 1/3
*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(b^2*d*f*x + a*b*d*f)*sqrt(b*d*f)*weierstrassZeta(4/3*(b^2*d
^2*e^2 + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2 - (b^2*c*d + a*b*d^2)*f*e)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 + (2
*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3 - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2*e -
3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)/(b^3*d^3*f^3), weierstrassPInverse(4/3*(b^2*d^2*e^2 + (b^2*c^2 - a*b*c*d + a^
2*d^2)*f^2 - (b^2*c*d + a*b*d^2)*f*e)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2
*b*c*d^2 + 2*a^3*d^3)*f^3 - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*f^2*e - 3*(b^3*c*d^2 + a*b^2*d^3)*f*e^2)
/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))))/((a*b^3*c*d - a^2*b^2*d^2)*f^2*x + (a^2*b^2
*c*d - a^3*b*d^2)*f^2 - ((b^4*c*d - a*b^3*d^2)*f*x + (a*b^3*c*d - a^2*b^2*d^2)*f)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x} \sqrt {e + f x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Integral(1/((a + b*x)**(3/2)*sqrt(c + d*x)*sqrt(e + f*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e + f*x)^(1/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)),x)

[Out]

int(1/((e + f*x)^(1/2)*(a + b*x)^(3/2)*(c + d*x)^(1/2)), x)

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