Optimal. Leaf size=543 \[ \frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac {e^{3/2} \sqrt {c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (-8 c^3 f^3+19 c^2 d e f^2-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (-8 c^3 f^3+19 c^2 d e f^2-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.64, antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {528, 531, 418, 492, 411} \[ \frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac {x \sqrt {c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt {e+f x^2}}+\frac {e^{3/2} \sqrt {c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 411
Rule 418
Rule 492
Rule 528
Rule 531
Rubi steps
\begin {align*} \int \left (a+b x^2\right ) \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2} \, dx &=\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac {\int \sqrt {c+d x^2} \sqrt {e+f x^2} \left (-(b c-7 a d) e+(3 b d e-4 b c f+7 a d f) x^2\right ) \, dx}{7 d}\\ &=\frac {(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d^2}+\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac {\int \frac {\sqrt {c+d x^2} \left (-e (4 b c (2 d e-c f)-7 a d (5 d e-c f))+\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x^2\right )}{\sqrt {e+f x^2}} \, dx}{35 d^2}\\ &=\frac {\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d^2 f}+\frac {(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d^2}+\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac {\int \frac {c e \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right )+\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d^2 f}\\ &=\frac {\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d^2 f}+\frac {(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d^2}+\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac {\left (c e \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d^2 f}+\frac {\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 d^2 f}\\ &=\frac {\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^3 f \sqrt {e+f x^2}}+\frac {\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d^2 f}+\frac {(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d^2}+\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac {e^{3/2} \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (e \left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{105 d^3 f}\\ &=\frac {\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d^3 f \sqrt {e+f x^2}}+\frac {\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 d^2 f}+\frac {(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 d^2}+\frac {b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}-\frac {\sqrt {e} \left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {e^{3/2} \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.71, size = 372, normalized size = 0.69 \[ \frac {i e \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (c f-d e) \left (b \left (4 c^2 f^2-6 c d e f+6 d^2 e^2\right )-7 a d f (c f+3 d e)\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+f x \left (-\sqrt {\frac {d}{c}}\right ) \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-7 a d f \left (c f+6 d e+3 d f x^2\right )+4 b c^2 f^2-3 b c d f \left (3 e+f x^2\right )-3 b d^2 \left (e^2+8 e f x^2+5 f^2 x^4\right )\right )-i e \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )+b \left (8 c^3 f^3-19 c^2 d e f^2+9 c d^2 e^2 f-6 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{105 c^2 f^2 \left (\frac {d}{c}\right )^{5/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b f x^{4} + {\left (b e + a f\right )} x^{2} + a e\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 1331, normalized size = 2.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b x^{2}\right ) \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________