3.1929 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^7) + (8*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(5/2))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^6) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(5/2))/(315*(c*d^2 - a*e^2)^3*(d + e*x)^5)

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Rubi [A]  time = 0.0780033, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {658, 650} \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^7) + (8*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(5/2))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^6) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(5/2))/(315*(c*d^2 - a*e^2)^3*(d + e*x)^5)

Rule 658

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

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), 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 + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^7} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac{(4 c d) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, dx}{9 \left (c d^2-a e^2\right )}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac{\left (8 c^2 d^2\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{63 \left (c d^2-a e^2\right )^2}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 \left (c d^2-a e^2\right )^3 (d+e x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0703643, size = 94, normalized size = 0.55 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 e^4-10 a c d e^2 (9 d+2 e x)+c^2 d^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )}{315 (d+e x)^7 \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(35*a^2*e^4 - 10*a*c*d*e^2*(9*d + 2*e*x) + c^2*d^2*(63*d^2 + 36*d*e*x + 8*e
^2*x^2)))/(315*(c*d^2 - a*e^2)^3*(d + e*x)^7)

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Maple [A]  time = 0.047, size = 146, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-20\,acd{e}^{3}x+36\,{c}^{2}{d}^{3}ex+35\,{a}^{2}{e}^{4}-90\,ac{d}^{2}{e}^{2}+63\,{c}^{2}{d}^{4} \right ) }{315\, \left ( ex+d \right ) ^{6} \left ({a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,a{c}^{2}{d}^{4}{e}^{2}-{c}^{3}{d}^{6} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-20*a*c*d*e^3*x+36*c^2*d^3*e*x+35*a^2*e^4-90*a*c*d^2*e^2+63*c^2*d^4)*(c*d
*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/(e*x+d)^6/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 89.4596, size = 937, normalized size = 5.48 \begin{align*} \frac{2 \,{\left (8 \, c^{4} d^{4} e^{2} x^{4} + 63 \, a^{2} c^{2} d^{4} e^{2} - 90 \, a^{3} c d^{2} e^{4} + 35 \, a^{4} e^{6} + 4 \,{\left (9 \, c^{4} d^{5} e - a c^{3} d^{3} e^{3}\right )} x^{3} + 3 \,{\left (21 \, c^{4} d^{6} - 6 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (63 \, a c^{3} d^{5} e - 72 \, a^{2} c^{2} d^{3} e^{3} + 25 \, a^{3} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{315 \,{\left (c^{3} d^{11} - 3 \, a c^{2} d^{9} e^{2} + 3 \, a^{2} c d^{7} e^{4} - a^{3} d^{5} e^{6} +{\left (c^{3} d^{6} e^{5} - 3 \, a c^{2} d^{4} e^{7} + 3 \, a^{2} c d^{2} e^{9} - a^{3} e^{11}\right )} x^{5} + 5 \,{\left (c^{3} d^{7} e^{4} - 3 \, a c^{2} d^{5} e^{6} + 3 \, a^{2} c d^{3} e^{8} - a^{3} d e^{10}\right )} x^{4} + 10 \,{\left (c^{3} d^{8} e^{3} - 3 \, a c^{2} d^{6} e^{5} + 3 \, a^{2} c d^{4} e^{7} - a^{3} d^{2} e^{9}\right )} x^{3} + 10 \,{\left (c^{3} d^{9} e^{2} - 3 \, a c^{2} d^{7} e^{4} + 3 \, a^{2} c d^{5} e^{6} - a^{3} d^{3} e^{8}\right )} x^{2} + 5 \,{\left (c^{3} d^{10} e - 3 \, a c^{2} d^{8} e^{3} + 3 \, a^{2} c d^{6} e^{5} - a^{3} d^{4} e^{7}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(8*c^4*d^4*e^2*x^4 + 63*a^2*c^2*d^4*e^2 - 90*a^3*c*d^2*e^4 + 35*a^4*e^6 + 4*(9*c^4*d^5*e - a*c^3*d^3*e^3
)*x^3 + 3*(21*c^4*d^6 - 6*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*x^2 + 2*(63*a*c^3*d^5*e - 72*a^2*c^2*d^3*e^3 + 25*a
^3*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^11 - 3*a*c^2*d^9*e^2 + 3*a^2*c*d^7*e^4 - a^3
*d^5*e^6 + (c^3*d^6*e^5 - 3*a*c^2*d^4*e^7 + 3*a^2*c*d^2*e^9 - a^3*e^11)*x^5 + 5*(c^3*d^7*e^4 - 3*a*c^2*d^5*e^6
+ 3*a^2*c*d^3*e^8 - a^3*d*e^10)*x^4 + 10*(c^3*d^8*e^3 - 3*a*c^2*d^6*e^5 + 3*a^2*c*d^4*e^7 - a^3*d^2*e^9)*x^3
+ 10*(c^3*d^9*e^2 - 3*a*c^2*d^7*e^4 + 3*a^2*c*d^5*e^6 - a^3*d^3*e^8)*x^2 + 5*(c^3*d^10*e - 3*a*c^2*d^8*e^3 + 3
*a^2*c*d^6*e^5 - a^3*d^4*e^7)*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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Giac [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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

Timed out