3.118 \(\int (d x)^m (a^2+2 a b x^3+b^2 x^6)^{5/2} \, dx\)

Optimal. Leaf size=313 \[ \frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^3\right )}+\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left (a+b x^3\right )}+\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+13}}{d^{13} (m+13) \left (a+b x^3\right )}+\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+16}}{d^{16} (m+16) \left (a+b x^3\right )}+\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]

[Out]

(a^5*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d*(1 + m)*(a + b*x^3)) + (5*a^4*b*(d*x)^(4 + m)*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(d^4*(4 + m)*(a + b*x^3)) + (10*a^3*b^2*(d*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
/(d^7*(7 + m)*(a + b*x^3)) + (10*a^2*b^3*(d*x)^(10 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^10*(10 + m)*(a + b
*x^3)) + (5*a*b^4*(d*x)^(13 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^13*(13 + m)*(a + b*x^3)) + (b^5*(d*x)^(16
+ m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^16*(16 + m)*(a + b*x^3))

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Rubi [A]  time = 0.137861, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1355, 270} \[ \frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^3\right )}+\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left (a+b x^3\right )}+\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+13}}{d^{13} (m+13) \left (a+b x^3\right )}+\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+16}}{d^{16} (m+16) \left (a+b x^3\right )}+\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(a^5*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d*(1 + m)*(a + b*x^3)) + (5*a^4*b*(d*x)^(4 + m)*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(d^4*(4 + m)*(a + b*x^3)) + (10*a^3*b^2*(d*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
/(d^7*(7 + m)*(a + b*x^3)) + (10*a^2*b^3*(d*x)^(10 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^10*(10 + m)*(a + b
*x^3)) + (5*a*b^4*(d*x)^(13 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^13*(13 + m)*(a + b*x^3)) + (b^5*(d*x)^(16
+ m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^16*(16 + m)*(a + b*x^3))

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int (d x)^m \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 (d x)^m+\frac{5 a^4 b^6 (d x)^{3+m}}{d^3}+\frac{10 a^3 b^7 (d x)^{6+m}}{d^6}+\frac{10 a^2 b^8 (d x)^{9+m}}{d^9}+\frac{5 a b^9 (d x)^{12+m}}{d^{12}}+\frac{b^{10} (d x)^{15+m}}{d^{15}}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{a^5 (d x)^{1+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d (1+m) \left (a+b x^3\right )}+\frac{5 a^4 b (d x)^{4+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^4 (4+m) \left (a+b x^3\right )}+\frac{10 a^3 b^2 (d x)^{7+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^7 (7+m) \left (a+b x^3\right )}+\frac{10 a^2 b^3 (d x)^{10+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^{10} (10+m) \left (a+b x^3\right )}+\frac{5 a b^4 (d x)^{13+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^{13} (13+m) \left (a+b x^3\right )}+\frac{b^5 (d x)^{16+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^{16} (16+m) \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0998909, size = 111, normalized size = 0.35 \[ \frac{x \left (\left (a+b x^3\right )^2\right )^{5/2} (d x)^m \left (\frac{10 a^2 b^3 x^9}{m+10}+\frac{10 a^3 b^2 x^6}{m+7}+\frac{5 a^4 b x^3}{m+4}+\frac{a^5}{m+1}+\frac{5 a b^4 x^{12}}{m+13}+\frac{b^5 x^{15}}{m+16}\right )}{\left (a+b x^3\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x*(d*x)^m*((a + b*x^3)^2)^(5/2)*(a^5/(1 + m) + (5*a^4*b*x^3)/(4 + m) + (10*a^3*b^2*x^6)/(7 + m) + (10*a^2*b^3
*x^9)/(10 + m) + (5*a*b^4*x^12)/(13 + m) + (b^5*x^15)/(16 + m)))/(a + b*x^3)^5

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Maple [A]  time = 0.007, size = 453, normalized size = 1.5 \begin{align*}{\frac{ \left ({b}^{5}{m}^{5}{x}^{15}+35\,{b}^{5}{m}^{4}{x}^{15}+445\,{b}^{5}{m}^{3}{x}^{15}+5\,a{b}^{4}{m}^{5}{x}^{12}+2485\,{b}^{5}{m}^{2}{x}^{15}+190\,a{b}^{4}{m}^{4}{x}^{12}+5714\,{b}^{5}m{x}^{15}+2555\,a{b}^{4}{m}^{3}{x}^{12}+3640\,{b}^{5}{x}^{15}+10\,{a}^{2}{b}^{3}{m}^{5}{x}^{9}+14810\,a{b}^{4}{m}^{2}{x}^{12}+410\,{a}^{2}{b}^{3}{m}^{4}{x}^{9}+34840\,a{b}^{4}m{x}^{12}+5950\,{a}^{2}{b}^{3}{m}^{3}{x}^{9}+22400\,a{b}^{4}{x}^{12}+10\,{a}^{3}{b}^{2}{m}^{5}{x}^{6}+36550\,{a}^{2}{b}^{3}{m}^{2}{x}^{9}+440\,{a}^{3}{b}^{2}{m}^{4}{x}^{6}+89240\,{a}^{2}{b}^{3}m{x}^{9}+6970\,{a}^{3}{b}^{2}{m}^{3}{x}^{6}+58240\,{a}^{2}{b}^{3}{x}^{9}+5\,{a}^{4}b{m}^{5}{x}^{3}+47260\,{a}^{3}{b}^{2}{m}^{2}{x}^{6}+235\,{a}^{4}b{m}^{4}{x}^{3}+123920\,{a}^{3}{b}^{2}m{x}^{6}+4085\,{a}^{4}b{m}^{3}{x}^{3}+83200\,{a}^{3}{b}^{2}{x}^{6}+{a}^{5}{m}^{5}+31685\,{a}^{4}b{m}^{2}{x}^{3}+50\,{a}^{5}{m}^{4}+100630\,{a}^{4}bm{x}^{3}+955\,{a}^{5}{m}^{3}+72800\,{a}^{4}b{x}^{3}+8650\,{a}^{5}{m}^{2}+36824\,{a}^{5}m+58240\,{a}^{5} \right ) x \left ( dx \right ) ^{m}}{ \left ( 1+m \right ) \left ( 4+m \right ) \left ( 7+m \right ) \left ( 10+m \right ) \left ( 13+m \right ) \left ( 16+m \right ) \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

x*(b^5*m^5*x^15+35*b^5*m^4*x^15+445*b^5*m^3*x^15+5*a*b^4*m^5*x^12+2485*b^5*m^2*x^15+190*a*b^4*m^4*x^12+5714*b^
5*m*x^15+2555*a*b^4*m^3*x^12+3640*b^5*x^15+10*a^2*b^3*m^5*x^9+14810*a*b^4*m^2*x^12+410*a^2*b^3*m^4*x^9+34840*a
*b^4*m*x^12+5950*a^2*b^3*m^3*x^9+22400*a*b^4*x^12+10*a^3*b^2*m^5*x^6+36550*a^2*b^3*m^2*x^9+440*a^3*b^2*m^4*x^6
+89240*a^2*b^3*m*x^9+6970*a^3*b^2*m^3*x^6+58240*a^2*b^3*x^9+5*a^4*b*m^5*x^3+47260*a^3*b^2*m^2*x^6+235*a^4*b*m^
4*x^3+123920*a^3*b^2*m*x^6+4085*a^4*b*m^3*x^3+83200*a^3*b^2*x^6+a^5*m^5+31685*a^4*b*m^2*x^3+50*a^5*m^4+100630*
a^4*b*m*x^3+955*a^5*m^3+72800*a^4*b*x^3+8650*a^5*m^2+36824*a^5*m+58240*a^5)*(d*x)^m*((b*x^3+a)^2)^(5/2)/(1+m)/
(4+m)/(7+m)/(10+m)/(13+m)/(16+m)/(b*x^3+a)^5

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Maxima [A]  time = 1.03573, size = 328, normalized size = 1.05 \begin{align*} \frac{{\left ({\left (m^{5} + 35 \, m^{4} + 445 \, m^{3} + 2485 \, m^{2} + 5714 \, m + 3640\right )} b^{5} d^{m} x^{16} + 5 \,{\left (m^{5} + 38 \, m^{4} + 511 \, m^{3} + 2962 \, m^{2} + 6968 \, m + 4480\right )} a b^{4} d^{m} x^{13} + 10 \,{\left (m^{5} + 41 \, m^{4} + 595 \, m^{3} + 3655 \, m^{2} + 8924 \, m + 5824\right )} a^{2} b^{3} d^{m} x^{10} + 10 \,{\left (m^{5} + 44 \, m^{4} + 697 \, m^{3} + 4726 \, m^{2} + 12392 \, m + 8320\right )} a^{3} b^{2} d^{m} x^{7} + 5 \,{\left (m^{5} + 47 \, m^{4} + 817 \, m^{3} + 6337 \, m^{2} + 20126 \, m + 14560\right )} a^{4} b d^{m} x^{4} +{\left (m^{5} + 50 \, m^{4} + 955 \, m^{3} + 8650 \, m^{2} + 36824 \, m + 58240\right )} a^{5} d^{m} x\right )} x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

((m^5 + 35*m^4 + 445*m^3 + 2485*m^2 + 5714*m + 3640)*b^5*d^m*x^16 + 5*(m^5 + 38*m^4 + 511*m^3 + 2962*m^2 + 696
8*m + 4480)*a*b^4*d^m*x^13 + 10*(m^5 + 41*m^4 + 595*m^3 + 3655*m^2 + 8924*m + 5824)*a^2*b^3*d^m*x^10 + 10*(m^5
+ 44*m^4 + 697*m^3 + 4726*m^2 + 12392*m + 8320)*a^3*b^2*d^m*x^7 + 5*(m^5 + 47*m^4 + 817*m^3 + 6337*m^2 + 2012
6*m + 14560)*a^4*b*d^m*x^4 + (m^5 + 50*m^4 + 955*m^3 + 8650*m^2 + 36824*m + 58240)*a^5*d^m*x)*x^m/(m^6 + 51*m^
5 + 1005*m^4 + 9605*m^3 + 45474*m^2 + 95064*m + 58240)

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Fricas [A]  time = 1.59833, size = 886, normalized size = 2.83 \begin{align*} \frac{{\left ({\left (b^{5} m^{5} + 35 \, b^{5} m^{4} + 445 \, b^{5} m^{3} + 2485 \, b^{5} m^{2} + 5714 \, b^{5} m + 3640 \, b^{5}\right )} x^{16} + 5 \,{\left (a b^{4} m^{5} + 38 \, a b^{4} m^{4} + 511 \, a b^{4} m^{3} + 2962 \, a b^{4} m^{2} + 6968 \, a b^{4} m + 4480 \, a b^{4}\right )} x^{13} + 10 \,{\left (a^{2} b^{3} m^{5} + 41 \, a^{2} b^{3} m^{4} + 595 \, a^{2} b^{3} m^{3} + 3655 \, a^{2} b^{3} m^{2} + 8924 \, a^{2} b^{3} m + 5824 \, a^{2} b^{3}\right )} x^{10} + 10 \,{\left (a^{3} b^{2} m^{5} + 44 \, a^{3} b^{2} m^{4} + 697 \, a^{3} b^{2} m^{3} + 4726 \, a^{3} b^{2} m^{2} + 12392 \, a^{3} b^{2} m + 8320 \, a^{3} b^{2}\right )} x^{7} + 5 \,{\left (a^{4} b m^{5} + 47 \, a^{4} b m^{4} + 817 \, a^{4} b m^{3} + 6337 \, a^{4} b m^{2} + 20126 \, a^{4} b m + 14560 \, a^{4} b\right )} x^{4} +{\left (a^{5} m^{5} + 50 \, a^{5} m^{4} + 955 \, a^{5} m^{3} + 8650 \, a^{5} m^{2} + 36824 \, a^{5} m + 58240 \, a^{5}\right )} x\right )} \left (d x\right )^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

((b^5*m^5 + 35*b^5*m^4 + 445*b^5*m^3 + 2485*b^5*m^2 + 5714*b^5*m + 3640*b^5)*x^16 + 5*(a*b^4*m^5 + 38*a*b^4*m^
4 + 511*a*b^4*m^3 + 2962*a*b^4*m^2 + 6968*a*b^4*m + 4480*a*b^4)*x^13 + 10*(a^2*b^3*m^5 + 41*a^2*b^3*m^4 + 595*
a^2*b^3*m^3 + 3655*a^2*b^3*m^2 + 8924*a^2*b^3*m + 5824*a^2*b^3)*x^10 + 10*(a^3*b^2*m^5 + 44*a^3*b^2*m^4 + 697*
a^3*b^2*m^3 + 4726*a^3*b^2*m^2 + 12392*a^3*b^2*m + 8320*a^3*b^2)*x^7 + 5*(a^4*b*m^5 + 47*a^4*b*m^4 + 817*a^4*b
*m^3 + 6337*a^4*b*m^2 + 20126*a^4*b*m + 14560*a^4*b)*x^4 + (a^5*m^5 + 50*a^5*m^4 + 955*a^5*m^3 + 8650*a^5*m^2
+ 36824*a^5*m + 58240*a^5)*x)*(d*x)^m/(m^6 + 51*m^5 + 1005*m^4 + 9605*m^3 + 45474*m^2 + 95064*m + 58240)

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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((d*x)**m*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.20121, size = 1215, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

((d*x)^m*b^5*m^5*x^16*sgn(b*x^3 + a) + 35*(d*x)^m*b^5*m^4*x^16*sgn(b*x^3 + a) + 445*(d*x)^m*b^5*m^3*x^16*sgn(b
*x^3 + a) + 5*(d*x)^m*a*b^4*m^5*x^13*sgn(b*x^3 + a) + 2485*(d*x)^m*b^5*m^2*x^16*sgn(b*x^3 + a) + 190*(d*x)^m*a
*b^4*m^4*x^13*sgn(b*x^3 + a) + 5714*(d*x)^m*b^5*m*x^16*sgn(b*x^3 + a) + 2555*(d*x)^m*a*b^4*m^3*x^13*sgn(b*x^3
+ a) + 3640*(d*x)^m*b^5*x^16*sgn(b*x^3 + a) + 10*(d*x)^m*a^2*b^3*m^5*x^10*sgn(b*x^3 + a) + 14810*(d*x)^m*a*b^4
*m^2*x^13*sgn(b*x^3 + a) + 410*(d*x)^m*a^2*b^3*m^4*x^10*sgn(b*x^3 + a) + 34840*(d*x)^m*a*b^4*m*x^13*sgn(b*x^3
+ a) + 5950*(d*x)^m*a^2*b^3*m^3*x^10*sgn(b*x^3 + a) + 22400*(d*x)^m*a*b^4*x^13*sgn(b*x^3 + a) + 10*(d*x)^m*a^3
*b^2*m^5*x^7*sgn(b*x^3 + a) + 36550*(d*x)^m*a^2*b^3*m^2*x^10*sgn(b*x^3 + a) + 440*(d*x)^m*a^3*b^2*m^4*x^7*sgn(
b*x^3 + a) + 89240*(d*x)^m*a^2*b^3*m*x^10*sgn(b*x^3 + a) + 6970*(d*x)^m*a^3*b^2*m^3*x^7*sgn(b*x^3 + a) + 58240
*(d*x)^m*a^2*b^3*x^10*sgn(b*x^3 + a) + 5*(d*x)^m*a^4*b*m^5*x^4*sgn(b*x^3 + a) + 47260*(d*x)^m*a^3*b^2*m^2*x^7*
sgn(b*x^3 + a) + 235*(d*x)^m*a^4*b*m^4*x^4*sgn(b*x^3 + a) + 123920*(d*x)^m*a^3*b^2*m*x^7*sgn(b*x^3 + a) + 4085
*(d*x)^m*a^4*b*m^3*x^4*sgn(b*x^3 + a) + 83200*(d*x)^m*a^3*b^2*x^7*sgn(b*x^3 + a) + (d*x)^m*a^5*m^5*x*sgn(b*x^3
+ a) + 31685*(d*x)^m*a^4*b*m^2*x^4*sgn(b*x^3 + a) + 50*(d*x)^m*a^5*m^4*x*sgn(b*x^3 + a) + 100630*(d*x)^m*a^4*
b*m*x^4*sgn(b*x^3 + a) + 955*(d*x)^m*a^5*m^3*x*sgn(b*x^3 + a) + 72800*(d*x)^m*a^4*b*x^4*sgn(b*x^3 + a) + 8650*
(d*x)^m*a^5*m^2*x*sgn(b*x^3 + a) + 36824*(d*x)^m*a^5*m*x*sgn(b*x^3 + a) + 58240*(d*x)^m*a^5*x*sgn(b*x^3 + a))/
(m^6 + 51*m^5 + 1005*m^4 + 9605*m^3 + 45474*m^2 + 95064*m + 58240)