3.6.91 \(\int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=133 \[ \frac {16 \sqrt {a+b x} \left (15 a^2 e^2-35 a b d e+23 b^2 d^2\right )}{15 \sqrt {d+e x} (b d-a e)^3}+\frac {6 d^2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac {8 d \sqrt {a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2} \]

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Rubi [A]  time = 0.13, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {949, 78, 37} \begin {gather*} \frac {16 \sqrt {a+b x} \left (15 a^2 e^2-35 a b d e+23 b^2 d^2\right )}{15 \sqrt {d+e x} (b d-a e)^3}+\frac {6 d^2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac {8 d \sqrt {a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]

[Out]

(6*d^2*Sqrt[a + b*x])/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (8*d*(8*b*d - 5*a*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^2*
(d + e*x)^(3/2)) + (16*(23*b^2*d^2 - 35*a*b*d*e + 15*a^2*e^2)*Sqrt[a + b*x])/(15*(b*d - a*e)^3*Sqrt[d + e*x])

Rule 37

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

Rule 78

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

Rule 949

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

Rubi steps

\begin {align*} \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx &=\frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {2 \int \frac {6 d (6 b d-5 a e)+20 e (b d-a e) x}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)}\\ &=\frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {8 d (8 b d-5 a e) \sqrt {a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac {\left (8 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^2}\\ &=\frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {8 d (8 b d-5 a e) \sqrt {a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac {16 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right ) \sqrt {a+b x}}{15 (b d-a e)^3 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 110, normalized size = 0.83 \begin {gather*} \frac {2 \sqrt {a+b x} \left (a^2 e^2 \left (149 d^2+260 d e x+120 e^2 x^2\right )-2 a b d e \left (175 d^2+306 d e x+140 e^2 x^2\right )+b^2 d^2 \left (225 d^2+400 d e x+184 e^2 x^2\right )\right )}{15 (d+e x)^{5/2} (b d-a e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]

[Out]

(2*Sqrt[a + b*x]*(a^2*e^2*(149*d^2 + 260*d*e*x + 120*e^2*x^2) - 2*a*b*d*e*(175*d^2 + 306*d*e*x + 140*e^2*x^2)
+ b^2*d^2*(225*d^2 + 400*d*e*x + 184*e^2*x^2)))/(15*(b*d - a*e)^3*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.14, size = 115, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt {a+b x} \left (120 a^2 e^2+\frac {9 d^2 e^2 (a+b x)^2}{(d+e x)^2}-\frac {50 b d^2 e (a+b x)}{d+e x}+\frac {20 a d e^2 (a+b x)}{d+e x}-300 a b d e+225 b^2 d^2\right )}{15 \sqrt {d+e x} (b d-a e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]

[Out]

(2*Sqrt[a + b*x]*(225*b^2*d^2 - 300*a*b*d*e + 120*a^2*e^2 + (9*d^2*e^2*(a + b*x)^2)/(d + e*x)^2 - (50*b*d^2*e*
(a + b*x))/(d + e*x) + (20*a*d*e^2*(a + b*x))/(d + e*x)))/(15*(b*d - a*e)^3*Sqrt[d + e*x])

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fricas [B]  time = 1.05, size = 293, normalized size = 2.20 \begin {gather*} \frac {2 \, {\left (225 \, b^{2} d^{4} - 350 \, a b d^{3} e + 149 \, a^{2} d^{2} e^{2} + 8 \, {\left (23 \, b^{2} d^{2} e^{2} - 35 \, a b d e^{3} + 15 \, a^{2} e^{4}\right )} x^{2} + 4 \, {\left (100 \, b^{2} d^{3} e - 153 \, a b d^{2} e^{2} + 65 \, a^{2} d e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{15 \, {\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} + {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/15*(225*b^2*d^4 - 350*a*b*d^3*e + 149*a^2*d^2*e^2 + 8*(23*b^2*d^2*e^2 - 35*a*b*d*e^3 + 15*a^2*e^4)*x^2 + 4*(
100*b^2*d^3*e - 153*a*b*d^2*e^2 + 65*a^2*d*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^3*d^6 - 3*a*b^2*d^5*e + 3*a^
2*b*d^4*e^2 - a^3*d^3*e^3 + (b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*x^3 + 3*(b^3*d^4*e^2 - 3
*a*b^2*d^3*e^3 + 3*a^2*b*d^2*e^4 - a^3*d*e^5)*x^2 + 3*(b^3*d^5*e - 3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2
*e^4)*x)

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giac [F(-1)]  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((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.01, size = 150, normalized size = 1.13 \begin {gather*} -\frac {2 \sqrt {b x +a}\, \left (120 a^{2} e^{4} x^{2}-280 a b d \,e^{3} x^{2}+184 b^{2} d^{2} e^{2} x^{2}+260 a^{2} d \,e^{3} x -612 a b \,d^{2} e^{2} x +400 b^{2} d^{3} e x +149 a^{2} d^{2} e^{2}-350 a b \,d^{3} e +225 b^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x)

[Out]

-2/15*(b*x+a)^(1/2)*(120*a^2*e^4*x^2-280*a*b*d*e^3*x^2+184*b^2*d^2*e^2*x^2+260*a^2*d*e^3*x-612*a*b*d^2*e^2*x+4
00*b^2*d^3*e*x+149*a^2*d^2*e^2-350*a*b*d^3*e+225*b^2*d^4)/(e*x+d)^(5/2)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b
^3*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 4.30, size = 268, normalized size = 2.02 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {x^2\,\left (240\,a^3\,e^4-40\,a^2\,b\,d\,e^3-856\,a\,b^2\,d^2\,e^2+800\,b^3\,d^3\,e\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\left (520\,a^3\,d\,e^3-926\,a^2\,b\,d^2\,e^2+100\,a\,b^2\,d^3\,e+450\,b^3\,d^4\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {2\,a\,d^2\,\left (149\,a^2\,e^2-350\,a\,b\,d\,e+225\,b^2\,d^2\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {16\,b\,x^3\,\left (15\,a^2\,e^2-35\,a\,b\,d\,e+23\,b^2\,d^2\right )}{15\,e\,{\left (a\,e-b\,d\right )}^3}\right )}{x^3\,\sqrt {a+b\,x}+\frac {d^3\,\sqrt {a+b\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {a+b\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {a+b\,x}}{e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((15*d^2 + 8*e^2*x^2 + 20*d*e*x)/((a + b*x)^(1/2)*(d + e*x)^(7/2)),x)

[Out]

-((d + e*x)^(1/2)*((x^2*(240*a^3*e^4 + 800*b^3*d^3*e - 856*a*b^2*d^2*e^2 - 40*a^2*b*d*e^3))/(15*e^3*(a*e - b*d
)^3) + (x*(450*b^3*d^4 + 520*a^3*d*e^3 - 926*a^2*b*d^2*e^2 + 100*a*b^2*d^3*e))/(15*e^3*(a*e - b*d)^3) + (2*a*d
^2*(149*a^2*e^2 + 225*b^2*d^2 - 350*a*b*d*e))/(15*e^3*(a*e - b*d)^3) + (16*b*x^3*(15*a^2*e^2 + 23*b^2*d^2 - 35
*a*b*d*e))/(15*e*(a*e - b*d)^3)))/(x^3*(a + b*x)^(1/2) + (d^3*(a + b*x)^(1/2))/e^3 + (3*d*x^2*(a + b*x)^(1/2))
/e + (3*d^2*x*(a + b*x)^(1/2))/e^2)

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sympy [F(-1)]  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((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(7/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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