3.21.57 \(\int \frac {(d+e x)^{7/2}}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [2057]

Optimal. Leaf size=233 \[ \frac {32 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 \sqrt {d+e x}}+\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3}+\frac {12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d} \]

[Out]

12/35*(-a*e^2+c*d^2)*(e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2+2/7*(e*x+d)^(5/2)*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d+32/35*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/(e*x
+d)^(1/2)+16/35*(-a*e^2+c*d^2)^2*(e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3

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Rubi [A]
time = 0.12, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \begin {gather*} \frac {32 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^4 d^4 \sqrt {d+e x}}+\frac {16 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^3 d^3}+\frac {12 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^4*d^4*Sqrt[d + e*x]) + (16*(c*d^2 - a
*e^2)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^3*d^3) + (12*(c*d^2 - a*e^2)*(d + e*x
)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2) + (2*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2])/(7*c*d)

Rule 662

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

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac {\left (6 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 d}\\ &=\frac {12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac {\left (24 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3}+\frac {12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 d^3}\\ &=\frac {32 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 \sqrt {d+e x}}+\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3}+\frac {12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 131, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (7 d+e x)-2 a c^2 d^2 e^2 \left (35 d^2+14 d e x+3 e^2 x^2\right )+c^3 d^3 \left (35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3\right )\right )}{35 c^4 d^4 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(7*d + e*x) - 2*a*c^2*d^2*e^2*(35*d^2 + 14*d*e*x
+ 3*e^2*x^2) + c^3*d^3*(35*d^3 + 35*d^2*e*x + 21*d*e^2*x^2 + 5*e^3*x^3)))/(35*c^4*d^4*Sqrt[d + e*x])

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Maple [A]
time = 0.73, size = 150, normalized size = 0.64

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-5 c^{3} d^{3} e^{3} x^{3}+6 a \,c^{2} d^{2} e^{4} x^{2}-21 c^{3} d^{4} e^{2} x^{2}-8 a^{2} c d \,e^{5} x +28 a \,c^{2} d^{3} e^{3} x -35 c^{3} d^{5} e x +16 e^{6} a^{3}-56 e^{4} d^{2} a^{2} c +70 d^{4} e^{2} c^{2} a -35 d^{6} c^{3}\right )}{35 \sqrt {e x +d}\, c^{4} d^{4}}\)	\(150\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-5 c^{3} d^{3} e^{3} x^{3}+6 a \,c^{2} d^{2} e^{4} x^{2}-21 c^{3} d^{4} e^{2} x^{2}-8 a^{2} c d \,e^{5} x +28 a \,c^{2} d^{3} e^{3} x -35 c^{3} d^{5} e x +16 e^{6} a^{3}-56 e^{4} d^{2} a^{2} c +70 d^{4} e^{2} c^{2} a -35 d^{6} c^{3}\right ) \sqrt {e x +d}}{35 c^{4} d^{4} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/35/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-5*c^3*d^3*e^3*x^3+6*a*c^2*d^2*e^4*x^2-21*c^3*d^4*e^2*x^2-8*a
^2*c*d*e^5*x+28*a*c^2*d^3*e^3*x-35*c^3*d^5*e*x+16*a^3*e^6-56*a^2*c*d^2*e^4+70*a*c^2*d^4*e^2-35*c^3*d^6)/c^4/d^
4

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Maxima [A]
time = 0.31, size = 184, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (5 \, c^{4} d^{4} x^{4} e^{3} + 35 \, a c^{3} d^{6} e - 70 \, a^{2} c^{2} d^{4} e^{3} + 56 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + {\left (21 \, c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{3} + {\left (35 \, c^{4} d^{6} e - 7 \, a c^{3} d^{4} e^{3} + 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (35 \, c^{4} d^{7} - 35 \, a c^{3} d^{5} e^{2} + 28 \, a^{2} c^{2} d^{3} e^{4} - 8 \, a^{3} c d e^{6}\right )} x\right )}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/35*(5*c^4*d^4*x^4*e^3 + 35*a*c^3*d^6*e - 70*a^2*c^2*d^4*e^3 + 56*a^3*c*d^2*e^5 - 16*a^4*e^7 + (21*c^4*d^5*e^
2 - a*c^3*d^3*e^4)*x^3 + (35*c^4*d^6*e - 7*a*c^3*d^4*e^3 + 2*a^2*c^2*d^2*e^5)*x^2 + (35*c^4*d^7 - 35*a*c^3*d^5
*e^2 + 28*a^2*c^2*d^3*e^4 - 8*a^3*c*d*e^6)*x)/(sqrt(c*d*x + a*e)*c^4*d^4)

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Fricas [A]
time = 3.87, size = 170, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (35 \, c^{3} d^{5} x e + 35 \, c^{3} d^{6} + 8 \, a^{2} c d x e^{5} - 16 \, a^{3} e^{6} - 2 \, {\left (3 \, a c^{2} d^{2} x^{2} - 28 \, a^{2} c d^{2}\right )} e^{4} + {\left (5 \, c^{3} d^{3} x^{3} - 28 \, a c^{2} d^{3} x\right )} e^{3} + 7 \, {\left (3 \, c^{3} d^{4} x^{2} - 10 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{35 \, {\left (c^{4} d^{4} x e + c^{4} d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/35*(35*c^3*d^5*x*e + 35*c^3*d^6 + 8*a^2*c*d*x*e^5 - 16*a^3*e^6 - 2*(3*a*c^2*d^2*x^2 - 28*a^2*c*d^2)*e^4 + (5
*c^3*d^3*x^3 - 28*a*c^2*d^3*x)*e^3 + 7*(3*c^3*d^4*x^2 - 10*a*c^2*d^4)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 +
a*d)*e)*sqrt(x*e + d)/(c^4*d^4*x*e + c^4*d^5)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3061 deep

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Giac [A]
time = 1.10, size = 389, normalized size = 1.67 \begin {gather*} \frac {2 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{4} d^{4}} - \frac {32 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} + 3 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-1\right )}}{35 \, c^{4} d^{4}} + \frac {2 \, {\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{4} e^{2} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d^{2} e^{4} + 21 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d^{2} e + 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 21 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-4\right )}}{35 \, c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

2*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*e^(-1)/(c^4*
d^4) - 32/35*(sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 + 3*sqrt(-c*d^2*e + a*e^
3)*a^2*c*d^2*e^4 - sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-1)/(c^4*d^4) + 2/35*(35*((x*e + d)*c*d*e - c*d^2*e + a*
e^3)^(3/2)*c^2*d^4*e^2 - 70*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c*d^2*e^4 + 21*((x*e + d)*c*d*e - c*d^
2*e + a*e^3)^(5/2)*c*d^2*e + 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 21*((x*e + d)*c*d*e - c*d^
2*e + a*e^3)^(5/2)*a*e^3 + 5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-4)/(c^4*d^4)

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Mupad [B]
time = 1.04, size = 194, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (32\,a^3\,e^6-112\,a^2\,c\,d^2\,e^4+140\,a\,c^2\,d^4\,e^2-70\,c^3\,d^6\right )}{35\,c^4\,d^4\,e}-\frac {2\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^4-28\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{35\,c^3\,d^3}-\frac {2\,e^2\,x^3\,\sqrt {d+e\,x}}{7\,c\,d}+\frac {6\,e\,x^2\,\left (2\,a\,e^2-7\,c\,d^2\right )\,\sqrt {d+e\,x}}{35\,c^2\,d^2}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(((d + e*x)^(1/2)*(32*a^3*e^6 - 70*c^3*d^6 + 140*a*c^2*d^4*e^2
- 112*a^2*c*d^2*e^4))/(35*c^4*d^4*e) - (2*x*(d + e*x)^(1/2)*(8*a^2*e^4 + 35*c^2*d^4 - 28*a*c*d^2*e^2))/(35*c^
3*d^3) - (2*e^2*x^3*(d + e*x)^(1/2))/(7*c*d) + (6*e*x^2*(2*a*e^2 - 7*c*d^2)*(d + e*x)^(1/2))/(35*c^2*d^2)))/(x
+ d/e)

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